^ 

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>.*^' 


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IMAGE  EVALUATION 
TEST  TARGET  (MT-3) 


1.0     !^l^  1^ 

w  lU   12.2 

Eii»  "^ 

lit 

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H]0tQgraphic 

Sdmces 

Carporatim 


39  WIST  MAIN  STRHT 

WltSTIR,N.Y.  MSM 

(7U)  t72-4S03 


CIHM/ICMH 

Microfiche 

Series. 


CIHfVI/ICIVIH 
Collection  de 
microfiches. 


Canadian  Instituta  for  Historical  IMicroraproductions  /  Institut  Canadian  de  microraproductiona  historiquaa 


:\ 


Tachnical  and  Bibliographic  Notas/Notas  tachniquat  at  bibliographiquaa 


Tha  Inttituta  has  attamptad  to  obtain  tha  baat 
original  copy  availabia  for  filming.  Faiituras  of  this 
copy  which  may  ba  bibliographically  uniqua. 
which  may  altar  any  of  tha  imagas  in  tha 
raproduction,  or  which  may  significantly  changa 
tha  uaual  mathod  of  filming,  ara  chackad  balow. 


D 
D 
D 
D 
D 

n 
n 

D 
D 


n 


Colourad  covars/ 
Couvartura  da  coulaur 

Covars  damagad/ 
Couvartura  andommagte 

Covars  rastorad  and/or  laminatad/ 
Couvartura  rastaurte  at/ou  palliculAa 

Covar  titia  missing/ 

La  titra  da  couvartura  manqua 

Colo  irad  maps/ 

Cartas  gAographiquas  an  coulaur 

Colourad  ink  (i.a.  othar  than  blua  or  black)/ 
Encra  da  coulaur  (i.a.  autra  qua  blaua  ou  noira) 

Colourad  platas  and/or  illustrations/ 
Planchas  at/ou  illustrations  an  coulaur 

Bound  with  othar  matarial/ 
RaliA  avac  d'autras  documants 

Tight  binding  may  causa  shadows  or  distortion 
along  intarior  margin/ 

La  re  liura  sarr6a  paut  causar  da  I'ombra  ou  da  la 
distortion  la  long  da  la  marga  intAriaura 

Blank  laavas  addad  during  rastoration  may 
appaar  within  tha  taxt.  Whanavar  possibia,  thasa 
hava  baan  omittad  from  filming/ 
11  sa  paut  qua  cartainas  pagas  blanchas  ajoutias 
lors  d'una  rastauration  apparaissant  dans  la  taxta. 
mais,  iorsqua  cala  Atait  possibia,  cas  pagas  n'ont 
pas  it6  filmAas. 


L'Institut  a  microfilm*  la  maillaur  axamplaira 
qu'il  lui  a  Ati  possibia  da  sa  procurar.  Las  details 
da  cat  axamplaira  qui  sont  paut-*tra  uniquas  du 
point  da  vua  bibliographiqua,  qui  pauvant  modifier 
una  imaga  raproduita,  ou  qui  pauvant  axiger  una 
modification  dans  la  mithoda  normala  da  filmage 
sont  indiqute  ci-dassous. 


I      I   Colourad  pagas/ 


Pagaa  da  coulaur 

Pagas  damagad/ 
Pagas  andommagias 


□    Pagas  rastorad  and/or  laminated/ 
Pagas  rastaurias  at/ou  pallicultes 

r~>  Pagas  discoloured,  itained  or  foxed/ 
i}l\   Pagas  dicolor^as,  tachatias  ou  piqudes 

□Pagas  datachad/ 
Pagas  d^tachtes 

QShowthrough/ 
Transparanca 

□    Quality  of  print  varies/ 
Qualiti  inigala  da  {'impression 

□    l''icludas  supplementary  material/ 
Comprend  du  material  suppiimantaire 


D 

[3 


Only  edition  available/ 
Seule  Edition  disponible 

Pages  wholly  or  partially  obscured  by  errata 
slips,  tissues,  etc..  have  been  refilmed  to 
ensure  the  best  possible  image/ 
Les  pagas  totalament  ou  partiallement 
obscurcies  par  un  feuillet  d'errata,  una  peiure, 
etc..  ont  M  filmies  A  nouveau  de  faqon  A 
obtenir  la  meilleure  imaga  possibia. 


Q^ 


Additional  comments:/ 
Commentairas  supplAmantaires: 


Wrinkled  pages  may  film  slightly  out  of  focus. 


This  item  is  filmed  at  the  reduction  ratio  checked  below/ 

Ca  document  est  film*  au  taux  de  reduction  indiqu*  ci-dessous. 


10X 

14X 

18X 

22X 

26X 

30X 

1 

1 

^ 

12X 

16X 

20X 

24X 

28X 

32X 

i 

%'//■'■' 


,!* 


^C' 


TIm  copy  fllm«d  h«r«  haa  b««n  r«produc«<l  thank* 
to  tha  ganaroaity  of: 

Sflminary  of  Quabcc 
Library  ;     ^/  . 


L'axamplaira  flimA  fut  raproduit  grflca  i  la 
ginAroaiti  da: 

SimiiMiira  d«  Quibac 
Bibliothkiua 


Tha  imagaa  appaaring  hara  ara  ttia  baat  quality 
poaaibia  conaidaring  tha  condition  and  lagibillty 
of  tha  original  copy  and  in  Icaaping  with  tha 
filming  contract  apacifieationa. 


Laa  in*agaa  auK/antaa  ont  4t*  raproduitaa  avac  la 
plua  grand  aoin,  compta  tanu  da  la  condition  at 
da  la  nattati  da  raxamplaira  fiimA,  at  ^n 
conformity  avac  laa  eonditiona  du  contrat  da 
filmaga. 


Original  copiaa  in  printad  papar  covara  ara  filmad 
baginning  with  tha  front  eovar  and  anding  on 
tha  laat  paga  with  a  printad  or  illuatratad  impraa- 
aion,  or  tha  bacic  covar  whan  appropriata.  All 
othar  original  copiaa  ara  filmad  baginning  on  tha 
firat  paga  with  a  printad  or  illuatratad  impraa- 
aion,  and  anding  on  tha  laat  paga  with  a  printad 
or  illuatratad  impraaaion. 


Laa  axamplairaa  originaux  dont  la  couvartura  an 
papiar  aat  imprim4a  aont  filmte  an  commanpant 
par  la  pramiar  plat  at  an  tarminant  aoit  par  la 
darni4ra  paga  qui  comporta  una  amprainta 
d'improaaion  ou  dllluatration,  aoit  par  la  aacond 
plat,  aalon  la  eaa.  Toua  laa  autraa  axamplairaa 
originaux  aont  filmto  an  commanpant  par  la 
pramiAra  paga  qui  comporta  una  amprainta 
dlmpra^aion  ou  dllluatration  at  an  tarminant  par 
la  damiira  paga  qui  comporta  una  taila 
amprainta. 


Tha  laat  racordad  frama  on  aach  microficha 
ahail  contain  tha  aymbol  — i^-  (moaning  "CON- 
TINUED"), or  tha  aymbol  y  (moaning  "END"), 
whichavar  appliaa. 


Un  daa  aymbdaa  auhranta  apparattra  aur  la 
damiAra  imaga  da  chaqua  microficha,  aalon  la 
eaa:  la  aymbola  — »>  aignifia  "A  SUIVRE",  la 
aymbola  y  aignifia  "FIN". 


Mapa,  plataa,  charta.  ate.,  may  ba  filmad  at 
diffarant  reduction  ratioa.  Thoaa  too  large  to  ba 
entirely  included  in  one  expoaure  ara  filmed 
beginning  in  the  upper  left  hand  comer,  left  to 
right  and  top  to  bottom,  aa  many  framaa  aa 
required.  The  following  diagrama  illuatrate  the 
method: 


Lea  cartea,  planchea.  tableaux,  etc..  peuvent  Atre 
filmte  i  dee  taux  da  rMuction  diffAranta. 
Loraque  la  document  eat  trop  grand  pour  Atra 
reproiduit  en  un  aaul  ciichA,  11  eat  fiimA  A  partir 
da  I'angle  aupAriaur  gauche,  de  geuche  k  droite, 
et  de  haut  en  baa,  en  prenant  le  nombre 
d'imegea  niceaaaira.  Lea  diagrammee  auivanta 
illuatrent  la  m^hoda. 


t 

2 

3 

1 

2 

3 

4 

8 

6 

iM 


i  ( '• 


Fi^. 


/ 


^'Zff  . 


A. 


m 


^:mwf^'t 


w^p^- 


m 


Ftp  4 


I 


tii  -Vii-fii 


A? 


I 


7^3 


OUTLINES 


■/ 


iio  - . 


or 


Rs* 


ASTRONOMY: 


-.-JW 


BT 


SIR  JOHN  F.  W.  HERSCHEL, 


MJL  D.C.L.  F.R.8.L.  k  B.  Hon.  M .R.LA.  F.R.A.S.  F.0.8, 


Vonrtipondant  9r  Honorarj  Membtr  of  th»  Imparltl,  Rojral,  and  Natiooil,  AcadcmlM  of  SciaiiM* 

of  Berlin,  Bruuelt,  CopenbafaD,  Ontlinfrn,  Haarlem,  Hataarhuielti  (0.  8.),  Modana, 

Napla^  Faria,  Fetertburg,  Stockholm,  Turin,  Vianna,  and  WublngtOD  (U.  S); 

Iba  Italian  and  HalTatic  Bociatieij 

Uia  Academiai,  Inttilutei,  te.,  of  Albany  (C.  8.),  Bolofna,  Catania,  Dijon,  Lanianne, 

Nantei,  Padua,  Palermo,  Rome,  Venice,  Utracht,  and  Wilna; 

Ua  FbUomatbie  Society  of  Faria ;  Atiatic  Society  of  Beof al ;  South  African  Lit.  aid  PhiU  Soeialy  | 

Literary  and  Hiilorlcal  Society  of  Quebec ;  HIttorical  Society  of  New  Tork ; 

Royal  Medico-Chlrnrgical  Sec, and  Inat.  of  Cifil  Engineera,  LondOB< 

-•.  Geographical  Soc  of  Berlin ;  Aatrooomical  and 

Mataorologlcal  Soc  of  Brititb  Qulana; 

lie.  kc.  kc 


NEW   EDITION.         /r.%'''^*^ 


WITH  NUHEROCS  PLATES  AND  WOOD-CUTS. 


•^t 


PHILADELPHIA: 

BLANCHARD    &    LEA. 

1860. 


// 


NOTE 

10 

THE    FOURTH    EDITION. 


^^^^^i^i^^i^*^^^*^^*^^*^^*^^^^^^ 


Several  alterations  and  additions  are  made  in  this  Edition,  be- 
sides  what  have  been  introduced  into  the  Third,  to  bring  it  up 
to  the  actual  state  of  astronomical  discovery.  The  elements  of 
four  new  planets  (Parthenope,  Egeria,  Victoria,  and  Irene)  have 
been  added,  and  improved  elements  of  Iris,  Metis,  Hebe,  and 
Hjgeia,  substituted  for  the  provisional  elements  before  given. 
The  remarkable  discovery  of  an  additional  ring  of  Saturn,  and 
the  curious  researches  of  M.  Peters  on  the  proper  motion  of 
Sirius,  with  several  minor  features,  are  noticed.  Where  such 
additions  ar^i  introduced  in  the  text,  they  are  indicated  by  being 
enclosed  io  bracketfl  [     ]. 


J.  F.  W.  Herschel. 


I<ni>4pa,   Aiuf    &    iH&l. 


(▼) 


^t 


PREFACE. 


^^^^^^^^«*M» 


The  work  here  offered  to  the  Public  is  based  upon  and  may  be 
considered  as  an  extension,  and,  it  is  hoped,  an  improvement  of 
a  treatise  on  the  same  subject,  forming  Part  43,  of  the  Cabinet 
Cyclopeedia,  published  in  the  year  1833.  Its  object  and  general 
character  are  sufficiently  stated  in  the  introductory  chapter  of 
that  volume,  here  reprinted  with  little  alteration ;  but  an  oppor- 
tunity having  been  afforded  me  by  the  Proprietors,  preparatory 
to  its  re-appearance  in  a  form  of  more  pretension,  I  have  gladly 
availed  myself  of  it,  not  only  to  correct  some  errors  which,  to 
my  regret,  subsisted  in  the  former  volume,  but  to  remodel  it  alto- 
gether (though  in  complete  accordance  with  its  original  design  as 
a  work  of  explanation)  \  to  introduce  much  new  matter  in  the 
earlier  portions  of  it ;  to  re-write,  upon  a  far  more  matured  and 
comprehensive  plan,  the  part  relating  to  the  lunar  and  planetary 
perturbations,  and  to  bring  the  su^^jects  of  sidereal  and  nebular 
astronomy  to  the  level  of  the  present  state  of  our  knowledge  in 
those  departments. 

The  chief  novelty  in  the  volume,  as  it  now  stands,  will  be  found 
in  the  manner  in  which  the  subject  of  Perturbations  is  treated. 
It  is  not  —  it  cannot  be  made  elementary^  in  the  sense  in  which 
that  word  is  understood  in  these  days  of  light  reading.  The 
chapters  devoted  to  it  must,  therefore,  be  considered  as  addressed 
to  a  class  of  readers  in  possession  of  somewhat  more  mathematical 
knowledge  than  those  who  will  find  the  rest  of  the  work  readily 

(yii) 


Till 


PREFACE. 


and  easily  accessible ;  to  readers  desirous  of  preparing  them- 
selves, by  the  possession  of  a  sort  of  carte  du  pays,  for  a  cam- 
paign in  tho  most  difficult,  but  at  the  same  time  the  most  attract- 
ive and  the  most  remunerative  of  all  the  applications  of  modern 
geometry.  More  especially  they  may  be  considered  as  addressed 
to  students  in  that  university,  where  the  "  Principia"  of  Newton 
is  not,  nor  ever  will  be,  put  aside  as  an  obsolete  book,  behind  the 
age;  and  where  the  grand,  though  rude  outlines  of  the  lunar 
theory,  as  delivered  in  the  eleventh  section  of  that  immortal 
work,  are  studied  less  for  the  sake  of  the  theory  itself  than  for 
the  spirit  of  far-reaching  thought,  superior  to  and  disencumbered 
of  technical  aids,  which  distinguishes  that  beyond  any  other  pro- 
duction of  the  human  intellect. 

In  delivering  a  rational  as  distinguished  from  a  technical  expo- 
sition of  this  subject,  however,  the  course  pursued  by  Newton  in 
the  section  of  the  Principia  alluded  to,  has  by  no  means  been 
servilely  followed.  As  regards  the  perturbations  of  the  nodes 
and  inclinations,  indeed,  nothing  equally  luminous  can  ever  be 
substituted  for  his  explanation.  But  as  respects  the  other  dis- 
turbances, the  point  of  view  chosen  by  Newton  has  been  aban- 
doned for  another,  which  it  is  somewhat  difficult  to  perceive  why 
he  did  not,  himself,  select.  By  a  different  resolution  of  the  dis- 
turbing forces  from  that  adopted  by  him,  and  by  the  aid  of  a  few 
obvious  conclusions  from  the  laws  of  elliptic  motion  which  would 
have  found  their  place,  naturally  and  consecutively,  as  corollaries 
of  the  seventeenth  proposition  of  his  first  book  (a  proposition 
which  seems  almost  to  have  been  prepared  with  a  special  view  to 
this  application),  the  momentary  change  of  place  of  the  upper 
focus  of  the  disturbed  ellipse  is  brought  distinctly  under  inspec- 
tion ;  and  a  clearness  of  conception  introduced  into  the  pertur- 
bations of  the  excentricities,  perihelia,  and  epochs,  which  the 
author  does  not  think  it  presumption  to  believe  can  be  obtained 
by  no  other  method,  and  which  certainly  is  not  obtained  by  that 
from  which  it  is  a  departure.  It  would  be  out  of  keeping  with 
the  rest  of  the  work  to  have  introduced  into  this  part  of  it  any 
algebraic  investigations ;  else  it  would  have  been  easy  to  show 
that  the  mode  of  procedure  here  followed  leads  direct,  and  by 


1  ^  ---: 


;;^;;- 


PREFACE. 


i^ 


steps  (for  the  subject)  of  the  most  elementary  character,  to  the 
general  formulie  for  these  perturbations,  delivered  by  Laplace 
in  the  Mecanique  Celeste.'  . 


.>if 


The  reader  will  find  one  class  of  the  lunar  ^nd  planetary  in- 
equalities handled  in  a  very  different  manner  from  that  in  which 
their  explanation  is  usually  presented.  It  comprehends  tho^e 
which  are  characterized  as  incident  on  the  epoch,  the  principal 
among  them  being  the  annual  and  secular  equations  of  the  moon, 
and  that  very  delicate  and  obscure  part  of  the  perturbational 
theory  (so  little  satisfactory  in  the  manner  in  which  it  emerges 
from  the  analytical  treatment  of  the  subject),  the  constant  or 
permanent  effect  of  the  disturbing  force  in  altering  the  disturbed 
orbit.  I  will  venture  to  hope  that  what  is  here  stated  will  tend 
to  remove  some  rather  generally  diffused  misapprehensions  as 
to  the  true  bearings  of  Newton's  explanation  of  the  annual 
equation.'^ 

If  proof  were  wanted  of  the  inexhaustible  fertility  of  astro- 
nomical science  in  points  of  novelty  and  interest,  it  would  suffice 
to  adduce  the  addition  to  the  list  of  members  of  our  system  of 
no  less  than  eight  new  planets  and  satellites  during  the  prepara- 
tion of  these  sheets  for  the  press.  Among  them  is  one  whose 
discovery  must  ever  be  regarded  as  one  of  the  noblest  triumphs 
of  theory.  In  the  account  here  given  of  this  discovery,  I  trust 
to  have  expressed  myself  with  complete  impartiality ;  and  in  the 
exposition  of  the  perturbative  action  on  Uranus,  by  which  the 
existence  and  situation  of  the  disturbing  planet  became  revealed 
to  us,  I  have  endeavoured,  in  pursuance  of  the  general  plan  of 
this  work,  rather  to  exhibit  a  rational  view  of  the  dynamical 
action,  than  to  convey  the  slightest  idea  of  the  conduct  of  those 
masterpieces  of  analytical  skill  which  the  researches  of  Messrs. 
Leverrier  and  Adams  exhibit. 

To  the  latter  of  these  eminent  geometers,  as  well  as  to  my 
excellent  and  esteemed  friend  the  Astronomer  Royal,  I  have  to 


'  Livre  ii.  chop.  viii.  art.  67. 

•  Priucipia,  lib.  i.  prop.  06,  cor.  6. 


X  PREFACE. 

return  my  best  thanks  for  communications  which  would  have 
effectually  relieved  some  doubts  I  at  one  period  entertained,  had 
I  not  succeeded  in  the  interim  in  getting  clear  of  them,  as  to  the 
compatibility/  of  my  views  on  the  subject  of  the  annual  equation 
already  alluded  to,  with  the  tenor  of  Newton's  account  of  it.  To 
my  valued  friend,  Professor  De  Morgan,  I  am  indebted  for  some 
most  ingenious  suggestions  on  the  subject  of  the  mistakes  com- 
mitted in  the  early  working  of  the  Julian  reformation  of  the 
calendar,  of  which  I  should  have  availed  myself,  had  it  not  ap- 
peared preferable,  on  mature  consideration,  lo  present  the  sub- 
ject in  its  simplest  form,  avoiding  altogether  entering  into  mi- 
nutiae of  chronological  discussion. 

J.  F.  W.  Herschel. 


1  ! 


Collingwood,  April  12,  1849. 


V  \ 


CONTENTS. 


Preface Pageri! — x 

Introduction 17 

PART  I. 

CHAPTER  I. 

General  notions.  Apparent  and  real  motions.  Shape  and  size  of  the  Earth. 
The  horizon  and  its  dip.  The  atmosphere.  Refraction.  Twilight.  Appear- 
ances resulting  from  diurnal  motion.  From  change  of  station  in  general. 
Parallactic  motions.  Terrestrial  parallax.  That  of  the  stars  insensible. 
First  step  towards  forming  an  idea  of  the  distance  of  the  stars.  Copemican 
view  of  the  Earth's  motion.  Relative  motion.  Motions  partly  real,  partly 
apparent.  Geocentric  astronomy,  or  ideal  reference  of  pheenomena  to  the 
Earth's  centre  as  a  common  couveutional  station 24 


CHAPTER  II. 

Terminology  and  elementary  geometrical  conceptions  and  relations.  Termino- 
logy relating  to  the  globe  of  the  Earth  —  to  the  celestial  sphere.  Celestial 
perspective 62 

CHAPTER  III. 

Of  the  nature  of  astronomical  instruments  and  observations  in  general.  Of 
sidereal  and  solar  time.  Of  the  measurements  of  time.  Clocks,  chronome- 
ters. Of  astronomical  measurements.  Principle  of  telescopic  sights  to 
increase  the  accuracy  of  pointing.  Simplest  application  of  this  principle. 
The  transit  instrument.  Of  the  measurement  of  angular  intervals.  Methods 
of  increasing  the  accuracy  of  reading.  The  vernier.  The  microscope.  Of 
the  mural  circle.  The  Meridian  circle.  Fixation  of  polar  and  horizontal 
points.  The  level,  plumb-line,  artificial  horizon.  Principle  of  coUimation. 
Collimators  of  Rittenhouse,  Kater,  and  Benzenberg.  Of  compound  instru- 
ments with  co-ordinate  circles.  The  equatorial,  altitude,  and  azimuth  instru- 
ment. Theodolite.  Of  the  sextant  and  reflecting  circle.  Principle  of  repe- 
tition. Of  micrometers.  Parallel  wire  micrometer.  Principle  of  the  dupli- 
cation of  images.  The  heliometer.  Double  refracting  eye-piece.  Variable 
prism  micrometer.     Of  the  position  micrometer 70 


xH 


CONTENTS. 


CHAPTER  IV. 

or    OXOOBAFHT.  i 

Of  the  figure  of  the  Earth.  Its  exact  dimensions.  Its  form  that  of  equilibrium 
modified  by  centrifugal  force.  Variation  of  gravity  on  its  surface.  Statical 
and  dynamical  measures  of  gravity.  The  pciidulum.  Gravity  to  a  spheroid. 
Other  effects  of  the  Earth's  rotation.  Trade  winds.  Determination  of  geo- 
graphical positions — of  latitudes — of  longitudes.  Conduct  of  a  trigonometri- 
cal survey.  Of  maps.  Projections  of  the  sphere.  Measurement  of  heights 
by  the  barometer 118 

CHAPTER  V. 

or  UBANOOBAPHT. 

Construction  of  celestial  maps  and  globes  by  observations  of  right  ascension  and 
declination.  Celestial  objects  distinguished  into  fixed  and  erratic.  Of  the 
constellations.  Natural  regions  in  the  heavens.  The  Milky  Way.  The  Zo- 
diac. Of  the  ecliptic.  Celestial  latitudes  and  longitudes.  Precession  of  the 
equinoxes.  Nutation.  Aberration.  Refraction.  Parallax.  Summary  view 
of  the  uranographical  corrections 161 


CHAPTER  VI. 

OP    THE    sun's    MOTION. 

Apparent  motion  of  the  sun  not  uniform.  Its  apparent  diameter  also  variable. 
Variation  of  its  distance  concluded.  Its  apparent  orbit  an  ellipse  about  the 
focus.  Law  of  the  angular  velocity.  Equable  description  of  areas.  Parallax 
of  the  Sun.  Its  distance  and  magnitude.  Copernican  explanation  of  the 
Sun's  apparent  motion.  Parallelism  of  the  Earth's  axis.  The  seasons.  Heat 
received  from  the  Sun  in  different  parts  of  the  orbit.  Mean  and  true  longi- 
tudes of  the  Sun.  Equation  of  the  centre.  Sidereal,  tropical,  and  anoma- 
listic years.  Physical  constitution  of  the  Sun.  Its  spots.  Faculse.  Probable 
nature  and  causo  of  the  spots.  Atmosphere  of  the  Sun.  Its  supposed  clouds. 
Temperature  at  its  surface.  Its  expenditure  of  heat.  Terrestrial  effects  of 
solar  radiation 185 


CHAPTER  VII. 

Of  the  Moon.  Its  sidereal  period.  Its  apparent  diameter.  Its  parallax,  dis- 
tance, and  real  diameter.  First  approximation  to  its  orbit.  An  ellipse  about 
the  Earth  in  the  focus.  Its  excentricity  and  inclination.  Motion  of  its  nodes 
and  apsides.  Of  occultations  and  solar  eclipses  generally.  Limits  within 
which  they  are  possible.  They  prove  the  Moon  to  be  an  opaque  solid.  Its 
light  derived  from  the  Sun.  Its  phases.  Synodic  revolution  or  lunar  month. 
Of  eclipses  more  particularly.     Their  phenomena.     Their  periodical  recur- 


^ 


CONTENTS. 


rence  Physical  constitution  of  the  Moon.  Its  mountains  and  other  super- 
fleial  features.  Indications  of  former  roloanio  activitj.  Its  atmosphere. 
Climate.  Radiation  of  heat  from  its  surface.  Rotation  on  its  own  axis. 
Libration.     Appearance  of  the  Earth  from  it 218 

CHAPTER  VIII. 

Of  terrestrial  gravity.  Of  the  law  of  universal  gravitation.  Paths  of  projec- 
tiles, apparent,  real.  The  Moon  retained  in  her  orbit  by  gravity.  Its  law  of 
diminution.  Laws  of  elliptic  motion.  Orbit  of  the  Earth  round  the  Sun  in 
accordance  with  these  laws.  Masses  of  the  Earth  and  Sun  compared. 
Density  of  the  Sun.  Force  of  gravity  at  its  surface.  Disturbing  effect  of  the 
Sun  on  the  Moon's  motion 288 


CHAPTER  IX. 

or    THK    SOLAR    STSTBH. 

Apparent  motions  of  the  planets.  Their  stations  and  retrogradations.  The  Sun 
their  natural  centre  of  motion.  Inferior  planets.  Their  phases,  periods,  etc. 
Dimensions  and  form  of  their  orbits.  Transits  across  the  Sun.  Superior 
planets.  Their  distances,  periods,  etc.  Kepler's  laws  and  their  interpreta- 
tion. Elliptic  elements  of  a  planet's  orbit.  Its  heliocentric  and  geocentric 
place.  Empirical  law  of  planetary  distances ;  violated  in  the  case  of  Nep- 
tune. The  ultra-zodiacal  planets.  Physical  peculiarities  observable  in  each 
of  the  planets ; 242 

CHAPTER  X. 

or    THE    BATELtlTKS. 

Of  the  Moon,  as  a  satellite  of  the  Earth.  General  proximity  of  satellites  to 
their  primaries,  and  consequent  subordination  of  their  motions.  Masses  of 
the  primaries  concluded  from  the  periods  of  their  satellites.  Maintenance  of 
Kepler's  laws  in  the  secondary  systems.  Of  Jupiter's  satellites.  Their 
eclipses,  etc.  Velocity  of  light  discovered  by  their  means.  Satellites  of 
Saturn— of  Uranus — of  Neptune 282 


CHAPTER  XL 

or  COMETS. 

Great  number  of  recorded  comets.  The  number  of  those  unrecorded  probably 
much  greater.  General  description  of  a  comet.  Comets  without  tails,  or  with 
more  than  one.  Their  extreme  tenuity.  Their  probable  structure.  Motions 
conformable  to  the  law  of  gravity.  Actual  dimensions  of  comets.  Periodical 
return  of  several.  Halley's  comet.  Other  ancient  comets  probably  periodic. 
2 


I  i 


XIV 


CONTENTS. 


Enoke's  comet  —  Biela's — Faye's — Lexell's — De  Yico's — Broraen's — Peter's. 
Great  comet  of  1848.  Its  probable  identity  with  Beveral  older  comets.  Qreat 
interest  at  present  attached  to  cometary  astronomy,  and  its  reasons.  Re- 
marks on  oometary  orbits  in  general 295 


PART  n. 

or  THE  PLANETARY  PERTURBATIONS. 
CHAPTER  XII. 

Subject  proponnded.  Problem  of  three  bodies.  Superposition  of  small  motions. 
Estimation  of  the  disturbing  force.  Its  geometrical  representation.  Nume- 
rical estimation  in  particular  cases.  Resolution  into  rectangular  components. 
Radial,  transversal,  and  orthogonal  disturbing  forces.  Normal  and  tangential. 
Their  characteristic  effects.  Effects  of  the  orthogonal  force.  Motion  of  the 
nodes.  ConditioLS  of  their  advance  and  recess.  Cases  of  an  exterior  planet 
disturbed  by  an  interior.  The  reverse  case.  In  every  case  the  node  of  the 
disturbed  orbit  recedes  on  the  plane  of  the  disturbing  on  an  average.  Com- 
bined effect  of  many  such  disturbances.  Motion  of  the  Moon's  nodes. 
Change  of  inclination.  Conditions  of  its  increase  and  diminution.  Average 
effect  in  a  whole  revolution.  Compensation  in  a  complete  revolution  of  the 
nodes.  Lagrange's  theorem  of  the  stability  of  the  inclinations  of  the  plane- 
tary orbits.  Change  of  obliquity  of  the  ecliptic.  Precession  of  the  equinoxes 
explained.     Nutation.     Principle  of  forced  vibrations 826 

CHAPTER  XIII. 

THEOBT   OF  THB  AXIS,    PKBIHELIA,   AND   KXOBNTBIOITIES. 

Variation  of  elements  in  general.  Distinction  between  periodic  and  secular 
variations.  Geometrical  expression  of  tangential  and  normal  forces.  Varia- 
tion of  the  Major  Axis  produced  only  by  the  tangential  force.  Lagrange's 
theorem  of  the  conservation  of  the  mean  distances  and  periods.  Theory  of 
the  Perihelia  and  Excentricities.  Geometrical  representation  of  their  mo- 
mentary variations.  Estimation  of  the  disturbing  forces  in  nearly  circular 
orbits.  Application  to  the  case  of  the  Moon.  Theory  of  the  lunar  apsides 
and  excentricity.  Experimental  illustration.  Application  of  the  foregoing 
principles  to  the  planetary  theory.  Compensation  in  orbits  very  nearly  cir- 
cular. Effects  of  ellipticity.  General  results.  Lagrange's  theorem  of  the 
stability  of  the  excentricities  854 


CHAPTER  XIV. 

Of  the  inequalities  independent  of  the  excentricities.     The  Moon's  variation  and 
parallactic  inequality.     Analogous  planetary  inequalities.     Three  cascn  of 


CONTENTS. 


vr 


planetary  perturbation  distinguished.  Of  inequalities  dependent  on  the  excen- 
tricities.  Long  inequality  of  Jupiter  and  Saturn.  Law  of  reciprocity  between 
the  periodical  variations  of  the  elements  of  both  planets.  Long  inequality  of 
the  Earth  and  Venus.  Variation  of  the  epoch.  Inequalities  incident  on  the 
epoch  affecting  the  mean  motion.  Laterpretation  of  the  constant  part  of  these 
inequalities.  Annual  equation  of  the  Moon.  Her  secular  acceleration.  Lunar 
inequalities  due  to  the  action  of  Venus.  Effect  of  the  spheroidal  figure  of  the 
Earth  and  other  planets  on  the  motions  of  their  satellites.  Of  the  tides. 
Masses  of  disturbing  bodies  deducible  from  the  perturbations  they  produce. 
Mass  of  the  Moon,  and  of  Jupiter's  satellites,  how  ascertained.  Perturbations 
of  Uranus  resulting  in  the  discovery  of  Neptune 887 


PART  III. 

OP   SIDEREAL  ASTRONOMY. 

CHAPTER  XV. 

Of  the  fixed  stars.  Their  classification  by  magnitudes.  Photometric  scale  of 
magnitudes.  Conventional  or  vulgar  scale.  Photometric  comparison  of  stars. 
Distribution  of  stars  over  the  heavens.  Of  the  Milky  Way  or  galaxy.  Its 
supposed  form  that  of  a  fiat  stratum  partially  subdivided.  Its  visible  course 
among  the  constellations.  Its  internal  structure.  Its  apparently  indefinite 
extent  in  certain  directions.  Of  the  distance  of  the  fixed  stars.  Their 
annual  parallax.  Parallactic  unit  of  sidereal  distance.  Effect  of  parallax 
analogous  to  that  of  aberration.  How  distinguished  from  it.  Detection  of 
parallax  by  meridional  observations.  Henderson's  application  to  a  Centauri. 
By  differential  observations.  Discoveries  of  Bessel  and  Struve.  List  of  stars 
in  which  parallax  has  been  detected.  Of  the  real  magnitudes  of  the  stars. 
Comparison  of  their  lights  with  that  of  the  Sun 439 


CHAPTER  XVL 

Variable  and  periodical  stars.  List  of  those  already  known.  Irregularities  in 
their  periods  and  lustre  when  brightest.  Irregular  and  temporary  stars. 
Ancient  Chinese  records  of  several.  Missing  stars.  Double  stars.  Their 
classification.  Specimens  of  each  class.  Binary  systems.  Rovo]"tion  round 
each  other.  Describe  elliptic  orbits  under  the  Newtonian  law  'f  gravity. 
Elements  of  orbits  of  several.  Actual  dimensions  of  their  orbits.  Coloured 
double  stars.  Pheanomenon  of  complementary  colours.  Sanguine  stars. 
Proper  motion  of  the  stars.  Partly  accounted  for  by  a  real  motion  of  the  Sun. 
Situation  of  the  solar  apex.  Agreement  of  southern  and  northern  stars  in 
giving  the  same  result.  Principles  on  which  the  investigation  of  the  solar 
motion  depends.  Absolute  velocity  of  the  Sun's  motion.  Supposed  revolution 
of  the  whole  sidereal  system  round  a  common  centre.  Systematic  parallax 
and  aberration.  Effect  of  the  motion  of  light  in  altering  the  apparent  period 
of  a  binary  star  467 


ZVl 


CONTENTS. 


CHAPTER  XVII. 

or  CL0STXBS  or  stars  and  MBBVtiE. 

Of  olastering  groups  of  stars.  Globalar  clusters.  Their  stability  dynbinically 
possible.  List  of  the  most  remarkable.  Classification  of  nebuloe  and  clusters. 
Their  distribution  orer  the  heavens.  Irregular  clusters.  ResoWability  of 
nebnIsB.  Theory  of  the  formation  of  clusters  by  nebulous  subsidence.  Of 
elliptic  nebulae.  That  of  Andromeda.  Annular  and  planetary  nebulae. 
Double  nebuliB.  Nebulous  stars.  Connection  of  nebulae  with  double  stars. 
Insulated  nebulae  of  forms  not  wholly  irregular.  Of  amorphous  nebulae. 
Their  law  of  distribution  marks  them  as  outliers  of  the  galaxy.  Nebulae  and 
nebulous  group  of  Orion— of  Argo — of  Sagittarius — of  Cygnus.  The  Magel- 
lanic clouds.  Singular  nebula  in  the  greater  of  them.  The  zodiacal  light. 
Shooting  stars ^ 498 

PART  IV. 

OP    THE    ACCOUNT    OP    TIME. 

CHAPTER  XVIII. 

Natural  units  of  time.  Relation  of  the  sidereal  to  the  solar  day  affected  by 
precession.  Incommensurability  of  the  day  and  year.  Its  inconvenience. 
How  obviated.  The  Julian  Calendar.  Irregularities  at  its  first  introduction. 
Reformed  by  Augustus.  Gregorian  reformation.  Solar  and  lunar  cycles. 
Indiction.  Julian  period.  Table  of  chronological  eras.  Rules  for  calculating 
the  days  elapsed  between  given  dates.     Equinoctial  time 528 

APPENDIX. 

I.  Lists  of  Northern  and  Southern  Stars,  with  their  approximate  Magni- 
tudes, on  the  Vulgar  and  Photometric  Scales 541 

II.  Synoptic  Table  of  the  Elements  of  the  Planetary  System 543 

ni.  Synoptic  Table  of  the  Elements  of  the  Orbits  of  the  Satellites,  so  far 

as  they  are  known 545 

rV.  Elements  of  Periodical  Comets  at  their  last  appearance 648 

Index 649 


*• 

i 


OUTLINES 


OF 


ASTRONOMY. 


I  ( 


INTRODUCTION. 


(1.)  Every  student  who  enters  upon  a  scientific  pursuit,  especially  if 
at  a  somewhat  advanced  period  of  life,  will  find  not  only  that  he  has 
much  to  learn,  but  much  also  to  unlearn.  Familiar  ohjects  and  events 
are  far  from  presenting  themselves  to  our  senses  in  that  aspect  and  with 
those  connections  under  which  science  requires  them  to  he  viewed,  and 
which  constitute  their  rational  explanation.  There  is,  therefore,  every 
reason  to  expect  that  those  objects  and  relations  which,  taken  together, 
constitute  the  subject  he  is  about  to  enter  upon  will  have  been  previously 
apprehended  by  him,  at  least  imperfectly,  because  much  has  hitherto 
escaped  his  notice  which  is  essential  to  its  right  understanding :  and  not 
only  so,  but  too  often  also  erroneously,  owing  to  mistaken  analogies,  and 
the  general  prevalence  of  vulgar  errors.  As  a  first  preparation,  therefore, 
for  the  course  he  is  about  to  commence,  he  must  loosen  his  hold  on  all 
crude  and  hastily  adopted  notions,  and  must  strengthen  himself,  by  some- 
thing of  an  effort  and  a  resolve,  for  the  unprejudiced  admission  of  any 
conclusion  which  shall  appear  to  be  supported  by  careful  observation  and 
logical  argument,  even  should  it  prove  of  a  nature  adverse  to  notions  he 
may  have  previously  formed  for  himself,  or  taken  up,  without  examina- 
tion, on  the  credit  of  others.  Such  an  effort  is,  in  fact,  a  commencement 
of  that  intellectual  discipline  which  forms  one  of  the  most  important  ends 
of  all  science.  It  is  the  first  movement  of  approach  towards  that  state  of 
mental  purity  which  alone  can  fit  us  for  a  full  and  steady  perception  of 
moral  beauty  as  well  as  physical  adaptation.  It  is  the  "euphrasy  and 
rue"  with  which  we  must  "  purge  our  sight"  before  we  can  receive  and 
contemplate  as  they  are  tha  lineaments  of  truth  and  nature. 

2  (17) 


18 


OUTLINES  OF  ASTRONOMT. 


(2.)  Then)  ii  no  Boience  which,  more  than  Mtronomy,  stancU  in  need 
of  such  a  preparation,  or  drawi  more  largely  on  that  intellectual  liberality 
which  is  ready  to  adopt  whatever  is  demonstrated,  or  concede  whatever  is 
rendered  highly  probable,  however  new  and  uncommon  the  poinds  of  view 
may  be  in  which  objects  the  most  familiar  may  thereby  become  placed. 
Almost  all  its  conclusions  stand  in  open  and  striking  contradiction  with 
those  of  superficial  and  vulgar  observation,  and  with  what  appears  to 
every  one,  until  he  has  understood  and  weighed  the  proofs  to  the  con- 
trary, the  most  positive  evidence  of  his  senses.  Thus,  the  earth  on  which 
he  stands,  and  which  has  served  for  ages  as  the  unshaken  foundation  of 
the  firmest  structures,  either  of  art  or  nature,  is  divested  by  the  astro- 
nomer of  its  attribute  of  fixity,  and  conceived  by  him  as  turning  swiftly 
on  its  centre,  and  at  the  same  time  moving  onwards  through  space  with 
great  rapidity.  The  sun  and  the  moon,  which  appear  to  untaught  eyes 
round  bodies  of  no  very  considerable  size,  become  enlarged  in  his  imagi- 
nation into  vast  globes,  —  the  one  approaching  in  magnitude  to  the  earth 
itself,  the  other  immensely  surpassing  it.  The  planets,  which  appear 
only  as  stars  somewhat  brighter  than  the  rest,  are  to  him  spacious,  elabo- 
rate, and  habitable  worlds ;  several  of  them  much  greater  and  far  more 
curiously  furnished  than  the  earth  he  inhabits,  as  there  aie  also  others 
less  so ',  and  the  stars  themselves,  properly  so  called,  which  to  ordinary 
apprehension  present  only  lucid  sparks  or  brilliant  atoms,  are  to  him  suns 
of  various  and  transcendent  glory  —  effulgent  centres  of  life  and  light  to 
myriads  of  unseen  worlds.  So  that  when,  after  dilating  his  thoughts  to 
comprehend  the  grandeur  of  those  ideas  his  calculations  have  called  up, 
and  exhausting  his  imagination  and  the  powers  of  his  language  to  devise 
similes  and  metaphors  illustrative  of  the  immensity  of  the  scale  on  which 
his  universe  is  constructed,  he  shrinks  back  to  his  native  sphere ;  he  finds 
It,  in  comparison,  a  mere  point;  so  lost  —  even  in  the  minute  system  to 
which  it  belongs — as  to  be  invisible  and  unuuspeoted  from  some  of  its 
principal  and  remoter  members. 

(3.)  There  is  hardly  any  thing  which  sets  in  a  stronger  light  the  inhe- 
rent power  of  truth  over  the  mind  of  man,  when  opposed  by  no  motives 
of  interest  or  passion,  than  the  perfect  readiness  with  which  all  these  con- 
clusions are  assented  to  as  soon  as  their  evidence  is  clearly  apprehended, 
and  the  tenacious  hold  they  acquire  over  our  belief  when  once  admitted. 
In  the  conduct,  therefore,  of  this  volume,  I  shall  take  it  for  granted  that 
the  reader  is  more  desirous  to  lenm  the  system  which  it  is  its  object  to 
teach  as  it  now  stands,  than  to  raise  or  revive  objections  against  it ;  and 
that,  in  short,  he  comes  to  the  task  with  a  willing  mind ;  an  assumption 
which  will  not  only  save  the  trouble  of  piling  argument  on  argument  to 


.:■  .1/ 


\^i     * 


U 


INTRODUCTION. 


19 


ooDTinoe  the  noeptioal,  but  will  greatly  facilitate  hia  actual  progrets;  inaa- 
much  as  he  will  find  it  at  once  easier  and  more  satis&ctory  to  pursue  from 
the  outset  a  straight  and  definite  path,  than  to  be  constantly  stepping 
aside,  invoWing  himself  in  perplexities  and  circuits,  which,  after  all,  can 
only  terminate  in  finding  himself  compelled  to  adopt  the  same  road. 

(4.)  The  method,  therefore,  we  propose  to  follow  in  this  work  is  neither 
strictly  the  analytic  nor  the  synthetic,  but  rather  such  a  combination  of 
both,  with  a  leaning  to  the  latter,  as  may  best  suit  with  a  didactic  com- 
position.  Its  object  is  not  to  convince  or  refute  opponents,  nor  to  inquire, 
under  the  semblance  of  an  assumed  ignorance,  for  principles  of  which  we 
are  all  the  time  in  full  possession  —  but  simply  to  teach  what  is  knoum. 
The  moderate  limit  of  a  single  volume,  to  which  it  will  be  confined,  and 
the  necessity  of  being  on  every  point,  within  that  limit,  rather  diffuse  and 
copious  in  explanation,  as  well  as  the  eminently  matured  and  ascertained 
character  of  the  science  itself,  render  this  course  both  practicable  and 
eligible.  Practicable,  because  there  is  now  no  danger  of  any  revolution 
in  astronomy,  like  those  which  are  daily  changing  the  features  of  the  less 
advanced  sciences,  supervening,  to  destroy  all  our  hypotheses,  and  throw 
our  statements  into  confusion.  Eligible,  because  the  space  to  be  bestowed, 
either  in  combating  refuted  systems,  or  in  leading  the  reader  forward  by 
slow  and  measured  steps  from  the  known  to  the  unknown,  may  be  more 
advantageously  devoted  to  such  explanatory  illustrations  as  will  impress 
on  him  a  familiar  and,  as  it  were,  a  practical  sense  of  the  sequence  of 
phenomena,  and  the  manner  in  which  they  are  produced.  We  shall  not, 
then,  reject  the  analytic  course  where  it  leads  more  easily  and  directly  to 
our  objects,  or  in  any  way  fetter  ourselves  by  a  rigid  adherence  to  method. 
Writing  only  to  be  understood,  and  to  communicate  as  much  information 
in  as  little  space  as  possible,  consistently  with  its  distinct  and  effectual 
communication,  no  sacrifice  can  be  afforded  to  system,  to  form,  or  to 
affectation. 

(5.)  We  shall  take  for  granted,  from  the  outset,  the  Oopemican  system 
of  the  world;  relying  on  the  easy,  obvious,  and  natural  explanation  it 
affords  of  all  the  phenomena  as  they  come  to  be  described,  to  impress  the 
studeni;  with  a  sense  of  its  truth,  without  either  the  formality  of  demon- 
stration or  the  superfluous  tedium  of  eulogy,  calling  to  mind  that  impor- 
tant remark  of  Bacon :  — "  Theoriarum  vires,  arota  et  quasi  se  mutuo 
sustinente  partium  adaptatione,  quft  quasi  in  orbem  cobaerent,  firmantur*;'' 

'  "  The  confirmation  of  theories  relies  on  the  compact  adaptation  of  their  parts,  by 
which,  like  those  of  an  arch  or  dome,  they  mutually  sustain  each  other,  and  form  a 
coherent  whole."  This  is  what  Dr.  Whewell  expressively  terms  the  cfiuiUente  o' 
iaductjom. 


-V 


OUTLINIB  0f  A8TR0K0MT. 


not  failing,  however,  to  point  out  to  the  reader,  m  oocui  m  offerB,  the 
oontTMt  which  ill  ii^peiior  limplieity  offisn  to  the  complication  of  other 
hypotbesM. 

(6.)  The  preliminaiy  knowledge  whioh  it  is  deeinble  that  the  student 
should  possess,  in  order  for  the  more  adyantageoos  pemsal  of  the  following 
pages,  consists  in  the  flimiliar  practice  of  decimal  and  seiagesimal  arith- 
metic; some  moderate  acquaintance  whh  geometry  and  trigonometry, 
both  plane  and  spherical;  the  elementary  principles  of  mechanics;  and 
enouf^  of  optics  to  understand  the  construction  and  use  of  the  telescope, 
and  some  other  of  the  simpler  instruments.  Of  oonne,  the  more  of  nuch 
knowledge  he  brings  to  the  perusal,  the  easier  will  be  his  progress,  and 
the  more  complete  the  information  gained ;  but  we  shall  endeavour  in 
every  case,  as  far  as  it  can  be  done  without  a  sacrifice  of  clearness,  and  of 
that  useftil  brevity  which  consists  in  the  absence  of  prolixity  and  epi- 
sode, to  render  what  we  have  to  say  as  independent  of  other  books  as 
possible. 

(7.)  After  all,  I  must  distinctly  caution  such  of  my  readers  as  may 
commence  and  terminate  their  astronomical  studies  witii  the  present  work 
(though  of  such,  —  at  least  in  the  latter  predicament, — I  trust  the  num- 
ber will  be  few),  that  its  utmost  pretension  is  to  place  them  on  the 
threshold  of  this  particular  wing  of  the  temple  of  Science,  or  rather  on 
an  eminenee  exterior  to  it,  whence  they  may  obtain  something  like  a 
general  notion  of  its  structure ;  or,  at  most,  to  give  those  who  may  wish 
to  enter  a  ground-plan  of  its  accesses,  and  vut  them  in  possession  of  the 
pass-word.  Admisaon  to  its  sanctuary,  uutl  to  the  privileges  and  feelings 
of  a  votary,  is  only  to  be  gained  by  one  mea/ia,  —  sound  and  sufficient 
knowledge  of  mathematieSf  the  great  instrument  of  all  exact  inquiry, 
without  which  no  man  can  ever  make  such  advances  in  this  or  any  other 
of  Ike  higher  dtpartmenis  of  science  as  can  entitle  him  to  form  an  inde- 
pendent opinion  on  any  subject  of  discussion  within  their  range.  It  is 
not  vrithf'  t  an  effort  that  those  who  possess  this  knowledge  can  commu- 
nioate  on  «nch  subjects  with  those  who  do  not,  and  adapt  their  language 
and  their  illustrations  to  the  necessities  of  such  an  intercourse.  T'of  (Coi- 
tions which  to  the  one  are  almost  identical  ire  theorems  of  im7><>ri  'oJ 
difficulty  to  the  otuer ;  nor  is  their  evidence  presented  in  the  sa  v^ .  ic 
the  mind  of  each.  In  teaching  such  propositions,  under  such  circum- 
stances, the  ap^t^l  has  to  be  made,  not  to  t^iO  pure  and  abstract  reason, 
but  to  the  I  ;nse  of  analogy-— to  practice  and  experience :  principles  and 
modes  of  actior^  havd  to  be  established  not  by  direct  argument  from 
acknowledged  8XiO)Uh,  but  by  continually  recurring  to  the  sources  from 
which  the  axioms  tueip'^^  .:vej  have  Iwcn  drawn;  viz.  examples;  that  is  to 


f-! 


1/ 


IMTRODUOTION. 


21 


Bay,  by  bringiDg  forward  and  dwelling  (n  simple  and  familiar  iniUneea  in 
which  the  same  principles  and  the  same  .  r  aiuiilor  modes  of  action  take 
place :  t!  j  erecting,  as  it  were,  in  eat  h  p^tiouUr  cam,  a  separate  induc- 
tion, and  constructing  at  each  step  a  little  Wy  of  nck/K^  to  meet  its 
exigencies.  The  difference  is  that  uf  pioueerii  g  road  tlinigh  an  uu- 
traversed  country  and  advancing  at  eotic  .ilong  a  bruad  and  beaten  high< 
way ;  that  is  to  say,  if  we  are  determined  to  moke  ourselveti  distinctly 
understood,  and  will  appeal  to  reaoon  at  all.  As  for  the  method  of  L.98er- 
tion,  or  a  direct  demand  on  the/at<A  of  the  student  (tkougl  'n  some  "^m- 
plez  oases  ind<  pettsable,  where  illustrative  explanation  mou  A  defeat  its 
own  err^  I  ;  1  4u  uning  tedious  and  burdensome  to  both  partie  it  is  one 
whi  \  1  e  tui  ' ' .  uer  willingly  adopt  nor  would  recommend  to  others. 

(/■  )  (>r\  the  ( vher  hand,  although  it  is  something  new  to  abandon  the 
road  of  .i  athematical  demonstration  in  the  treatment  of  subjects  sui-  epti- 
1 .6  of  it,  and  to  teach  any  considerable  branch  of  science  entirely  or  ci  lefly 
by  th(;  way  of  illustration  and  familiar  parallels,  it  is  yet  not  impoeHiMo 
thai  those  who  are  already  well  acquainted  with  our  subject,  and  whose 
knowledge  has  been  acquired  by  that  confessedly  higher  practice  which  i 
incompatible  with  the  avowed  objects  of  the  present  work,  may  yet  find  their 
account  in  its  perusal,  —  for  this  reason,  that  it  is  always  of  advantage  to 
present  any  given  body  of  knowledge  to  the  mind  in  as  great  a  variety  of 
different  lights  as  possible.  It  is  a  property  of  illustrations  of  this  kind 
to  strike  no  two  minds  in  the  same  manner,  or  with  the  same  force; 
because  uo  two  minds  are  stored  with  the  same  images,  or  have  acquired 
their  notions  of  them  by  similar  habits.  Accordingly,  it  may  very  well 
happen,  that  a  proposition,  even  to  one  best  acquainted  with  it,  may  be 
placed  not  merely  in  a  new  and  uncommon,  but  in  a  more  impressive  and 
satisfactory  light  by  such  a  course  —  some  obscurity  may  be  dissipated, 
some  inward  misgivings  cleared  up,  or  even  some  links  supplied  which 
may  lead  to  the  perception  of  connections  and  deductions  altogether 
unknown  before.  And  the  probability  of  this  is  increased  when,  as  in 
the  present  instance,  tke  illustrations  chosen  have  not  been  studiously 
rclected  from  books,  but  are  such  as  have  presented  themselves  freely  to 
the  ttutuor's  mind  aa  being  most  in  harmony  with  his  own  views  -,  by 
which,  of  course,  he  means  to  lay  no  claim  to  originality  in  all  or  any  of 
them  beyond  what  thoj  may  really  possessv 

(9.)  Besides,  there  are  cases  in  the  application  of  mechanical  principles 
with  which  the  luttihenuUical  student  is  but  too  familiar,  where,  when  tho 
data  are  before  him,  ami  the  numerical  and  geometrical  relations  of  his 
problems  all  clear  to  bis  conception, — when  his  forces  are  estimated  and 
his  lines  measured,  —  HMy,  wheu  even  he  has  followed  up  the  application 


X 


y 


22 


OUTLINES   OF  ASTRONOMY. 


of  his  technical  processes,  and  fairly  arrived  at  his  conolasion,  —  there  is 
still  something  wanting  in  his  mind  —  not  in  the  evidence,  for  he  has 
examined  each  link,  and  finds  the  chain  complete  —  not  in  the  principlcSj 
for  those  he  well  knows  are  too  firmly  established  to  be  shaken  —  but  pre- 
cisely in  the  mode  of  action.  He  has  followed  out  a  train  of  reasoning 
by  logical  and  technical  rules,  but  the  signs  he  has  employed  are  not 
pictures  of  nature,  or  have  lost  their  original  meaning  as  such  to  his 
mind :  he  has  not  seen,  as  it  were,  the  process  of  nature  passing  under 
his  eye  in  an  instant  of  time,  and  presented  as  a  consecutive  whole  to  his 
imagination.  A  familiar  parallel,  or  an  illustration  drawn  from  some 
artificial  or  natural  process,  of  which  he  has  that  direct  and  individual 
impression  which  gives  it  a  reality  and  associates  it  with  a  name,  will,  in 
almost  every  such  case,  supply  in  a  moment  this  deficient  feature,  will 
convert  all  his  symbols  into  real  pictures,  and  infuse  an  animated  meaning 
into  what  was  before  a  lifeless  succession  of  words  and  signs.  I  cannot, 
indeed,  always  promise  myself  to  attain  this  degree  of  vividness  of  illus- 
tration, nor  are  the  points  to  be  elucidated  themselves  always  capable  of 
being  so  paraphrased  (if  I  may  use  the  expression)  by  any  single  in- 
stance adducible  in  the  ordinary  course  of  experience ;  but  the  object  will 
at  least  be  kept  in  view ;  and,  as  I  am  very  conscious  of  having,  in  making 
such  attempts,  gained  for  myself  much  clearer  views  of  several  of  the 
more  concealed  effects  of  planetary  perturbation  than  I  had  acquired  by 
their  mathematical  investigation  in  detail,  it  may  reasonably  be  hoped 
that  the  endeavour  will  not  always  be  unattended  with  a  similar  success 
in  others. 

(10.)  From  what  has  been  said,  it  will  be  evident  that  our  aim  is  not 
to  offer  to  the  public  a  technical  treatise,  in  which  the  student  of  practical 
or  theoretical  astronomy  shall  find  consigned  the  minute  description  of 
methods  of  observation,  or  the  formulae  he  requires  prepared  to  his  hand, 
or  their  demonstrations  drawn  out  in  detail.  In  all  these  the  present 
work  will  be  found  meagre,  and  quite  inadequate  to  his  wants.  Its  aim 
is  entirely  different;  being  to  present  in  each  case  the  mere  ultimate 
rationale  of  fact43,  arguments,  and  processes ;  and,  in  all  cases  of  mathe- 
matical application,  avoiding  whatever  would  tend  to  encumber  its  pages 
with  algebraic  or  geometrical  symbols,  to  place  under  his  inspection  that 
central  thread  of  common  sense  on  which  the  pearls  of  analytical  research 
are  invariably  strung ;  but  which,  by  the  attention  the  latter  claim  for 
themselves,  is  often  concealed  from  the  eye  of  the  gazer,  and  not  always 
disposed  in  the  straightest  and  most  convenient  form  to  follow  by  those 
who  string  them.  This  is  no  fault  of  those  who  have  conducted  the 
inquiries  to  which  we  allude.     The  contention  of  mind  for  which  they  call 


INTRODUCTION. 


2» 


is  enormous;  and  it  may,  perhaps,  be  owing  to  their  experience  of  how 
little  can  be  accompluhed  in  carrying  such  processes  on  to  their  conclu- 
sion, by  mere  ordinary  clearness  of  head;  and  how  necessary  it  often  is 
to  pay  more  attention  to  the  purely  mathematical  conditions  which  ensure 
success,  —  the  hooks^ind-eyes  of  their  equations  and  series,  —  than  to 
those  which  enchain  causes  with  their  effects,  and  both  with  the  human 
reason,  —  that  we  must  attribute  something  of  that  indistinctness  of  view 
which  is  often  complained  of  as  a  grievance  by  the  earnest  student,  and 
still  more  commonly  ascribed  ironically  to  the  native  oloudiness  of  an 
atmosphere  too  sublime  for  vulgar  comprehension.  We  think  we  shall 
render  good  service  to  both  classes  of  readers,  by  dissipating,  so  far  as  lies 
in  our  power,  that  accidental  obscurity,  and  by  showing  ordinary  untu- 
tored comprehension  clearly  what  it  can,  and  what  it  cannot,  hope  to 
attun. 

»     --  •:  t  1  ^  -  -  ,-fii^?.4,  'Vtu  '<•>  It  h  ^ 

.,*-^,    t     iiiK-i.    '^'"A^'^hM"      €^  ^Hi-iL  ^'.'m  .i%h*^-    /.»  v*f!'V.    ^    ',    -'(^t 


\ 


'*«    Its     Ui.^   ^ji'^n  .>",,'!>  A"*        fef*-i. 


,  fr  Ti"  -i 


.'      «''Jsll  .fp 


\y   ,^-«'/l.H 


"!     * 


4)    ', 


:!*'T 


■  r 

>  f 


ii-> 


WiSi 


«,. '  I 


-T^-R 


* 


OUTLINES  or  ASTKOKOMT. 


CHAPTER  I    ?5.  1^^  1    ' 


M«  -r- 


I  i;  >f ;'  i^ 


QENiaiAL  NOTIONS. — APPARENT  AND  REAL  MOTIONS.  —  SHAPE  AND 
SIZE  OP  THE  EARTH.  —  THE  HORIZON  AND  ITS  DIP.  —THE  ATMO- 
SPHERE. —  REFRACTION.  —  TWILIGHT.  —  APPEARANCES  RESULTING 
FROM  DITTRNAL  MOTION.  —  PROM  CHANGE  OP  STATION  IN  GENERAL. 
—  PARALIiAOTIO  MOTIONS. — TERRESTRIAL  PARALLAX. — THAT  OF 
THE  STARS  INSENSIBLE.  —  FIRST  STEP  TOWARDS  FORMING  AN  IDEA 
OP  THE  DISTANCE  OF  THE  STARS.  —  COPERNICAN  VIEW  OF  THE 
earth's  motion. — RELATIVE  MOTION. — MOTIONS  PARTLY  REAL, 
PARTLY  APPARENT. — GEOCENTRIC  ASTRONOMY,  OR  IDEAL  REFERENCE 
OF  PHJENOMENA  TO  THE  EARTH'S  CENTRE  AS  A  COMMON  CONVEN- 
TIONAL  STATION. 


(11.)  The  magnitudes,  distances,  arrangement,  and  motions  of  the 
great  bodies  which  make  up  the  visible  universe,  their  constitution  and 
physical  condition,  so  far  as  they  can  be  known  to  us,  with  their  mutual 
influences  and  actions  on  each  other,  so  far  as  they  can  be  traced  by  the 
effects  produced,  and  established  by  legitimate  reasoning,  form  the  assem- 
blage of  objects  to  which  the  intention  of  the  astronomer  is  directed. 
The  term  astronomy'  itself,  which  denotes  the  law  or  rule  of  the  astra  (by 
which  the  ancients  understood  not  only  the  stars  properly  so  called,  but 
the  sun,  the  moon,  and  all  the  visible  constituents  of  the  heavens),  suffi- 
ciently indicates  this ;  and,  although  the  term  astrology,  which  denotes  the 
reason,  theory,  or  interpretation  of  the  stars,*  has  become  degraded  in  its 
application,  and  confined  to  superstitious  and  delusive  attempts  to  divine 
future  events  by  their  dependence  on  pretended  planetary  influences,  the 
same  meaning  originally  attached  itself  to  that  epithet. 

(12.)  But,  besides  the  stars  and  other  celestial  bodies,  the  earth  itself, 
regarded  as  an  individual  body,  is  one  principal  object  of  the  astronomer's 
consideration,  and  indeed,  the  chief  of  all.     It  derives  its  importance,  in 

'  Aartip,  a  $lar  ;  yo/iof,  a  lava  ;  or  vc^tiv,  to  tend,  as  a  shepherd,  his  flock ;  so  that 
aarpovufios  means  "  shepherd  of  the  stars."  The  two  two  etymologies  are,  however, 
coincident . 

'  Aoyos,  reaton,  Of  a  word,  the  vehicle  of  reason ;  the  interpreter  of  thought. 


GENERAL   NOTIONS. ^^'> 


a  practical  as  well  as  theoretical  sense,  not  only  from  its  proximity,  and 
its  relation  to  us  as  animated  beings,  who  draw  from  it  the  supply  of  all 
our  wants,  but  as  the  station  from  which  we  see  all  the  rest,  and  as  the 
only  one  among  them  to  which  we  can,  in  the  &rst  instance,  refer  for  any 
determinate  marks  and  measures  by  which  to  recognize  their  changes  of 
situation,  or  with  which  to  compare  their  distances. 

(13.)  To  the  reader  who  now  for  the  first  time  takes  up  a  book  on 
astronomy,  it  will  na  doubt  seem  strange  to  class  the  earth  with  the 
heavenly  bodies,  and  to  assume  any  community  of  nature  among  things 
apparently  so  different.  For  what,  in  fact,  can  be  more  apparently  differ- 
ent than  the  vast  and  seemingly  immeasurable  extent  of  Axo  earth,  and  the 
stars  ?  The  earth  is  a  dark  and  opaque,  while  the  celestial  bodies  are 
brilliant.  We  perceive  in  it  no  n.  otion,  while  in  them  we  observe  a  con- 
tinual change  of  place,  as  we  view  them  at  different  hours  of  the  day  or 
night,  or  at  different  seasons  of  the  year.  The  ancients,  accordingly,  one 
or  two  of  the  more  enlightened  of  them  only  excepted,  admitted  no  such 
community  of  nature ;  and,  by  thus  placing  the  heavenly  bodies  and  their 
movements  without  the  pale  of  analogy  and  experience,  effectually  inter- 
cepted the  progress  of  all  reasoning  from  what  passes  here  below,  to  what 
is  going  on  in  the  regions  where  they  exist  and  move.  Under  such  con- 
ventions, astronomy,  as  a  science  of  cause  and  effect,  could  not  exist,  but 
must  be  limited  to  a  mere  registry  of  appearances,  unconnected  with  any 
attempt  to  account  for  them  on  reasonable  principles,  however  successful 
to  a  certain  extent  might  be  the  attempt  to  follow  out  their  order  of 
sequence,  and  to  establish  empirical  laws  expressive  of  this  order.  To 
get  rid  of  this  prejudice,  therefore,  is  the  first  step  towards  acquiring  a 
knowledge  of  what  is  really  the  case  j  and  the  student  has  made  his  first 
effort  towards  the  acquisition  of  sound  knowledge,  when  he  has  learnt  to 
familiarize  himself  with  the  idea  that  the  earth,  after  all,  may  be  nothing 
but  a  great  star.  How  correct  such  an  idea  may  be,  and  with  what  limi- 
tations and  modifications  it  is  to  be  admitted,  we  shall  see  presently.' 

(14.)  It  is  evident,  that,  to  form  any  just  notions  of  the  arrangement, 
in  space,  of  a  number  of  objects  which  we  cannot  approach  and  examine, 
but  of  which  all  the  information  we  can  gain  is  by  sitting  still  and  watch- 
ing their  evolutions,  it  must  be  very  important  for  us  to  know,  in  the  first 
instance,  whether  what  we  call  sitting  still  is  really  such :  whether  the 
station  from  which  we  view  them,  with  ourselves,  and  all  objects  which 
immediately  surround  us,  be  not  itself  in  motion,  unperceived  by  us ;  and 
if  so,  of  what  nature  that  motion  is.  The  apparent  places  of  a  number 
of  objects,  and  their  apparent  arrangement  with  respect  to  each  other, 
will  of  course  be  materially  dependent  on  the  situation  of  the  spectator 


/ 


/ 


Sd 


OUTLINES   OF  ASTRONOMY. 


among  them ;  and  if  this  situation  be  liable  to  change,  unknown  to  the 
spectator  himself,  an  appearance  of  change  in  the  respective  situations  of 
the  objects  will  arise,  without  the  reality.  If,  then,  such  be  actually  the 
case,  it  will  follow  that  all  the  movements  we  think  we  perceive  among 
the  stars  will  not  be  real  movements,  but  that  some  part,  at  least,  of 
whatever  changes  of  relative  place  we  perceive  among  them  must  be 
merely  apparent,  the  results  of  the  shifting  of  our  own  point  of  view ; 
and  that,  if  we  would  ever  arrive  at  a  knowledge  of  their  real  motions,  it 
can  only  be  by  first  investigating  our  own,  and  making  due  allowance  for 
its  effects.  Thus,  the  question  whether  the  earth  is  in  motion  or  at  rest, 
and  if  in  motion,  what  that  motion  is,  is  no  idle  inquiry,  but  one  on  which 
depends  our  only  chance  of  arriving  at  true  conclusions  respecting  the 
constitution  of  the  universe. 

(15.)  Nor  let  it  be  thought  strange  that  we  should  speak  of  a  motion 
existing  in  the  earth,  unperceived  by  its  inhabitants ;  we  must  remember 
that  it  is  of  the  earth  az  a  tohole,  with  all  that  it  holds  within  its  substance 
or  sustains  on  its  surface,  that  we  are  speaking ;  of  a  motion  common  to 
the  solid  mass  beneath,  to  the  ocean  which  flows  around  it,  the  air  that 
rests  upon  it,  and  the  clouds  which  float  above  it  in  the  air.  Such  a 
motion,  which  should  displace  no  terrestrial  object  from  its  relative  situa- 
tion among  others,  interfere  with  no  natural  processes,  and  produce  no 
sensations  of  shocks  or  jerks,  might,  it  is  very  evident,  subsist  undetected 
by  us.  There  is  no  peculiar  sensation  which  advertises  us  that  we  are  in 
motion.  We  perceive  jerks,  or  shocks,  it  is  true,  because  these  are  sud- 
den changes  of  motion,  produced,  as  the  laws  of  mechanics  teach  us,  by 
sudden  and  powerful  forces  acting  during  short  times ;  and  those  forces, 
applied  to  our  bodies,  are  what  we  feel.  When,  for  example,  we  are 
carried  along  in  a  carriage  with  the  blinds  down,  or  with  our  eyes  closed 
(to  keep  us  from  seeing  external  objects),  wo  perceive  a  tremor  arising  from 
inequalities  in  the  road,  over  which  the  carriage  is  successively  lifted  and 
let  fall,  but  we  have  no  sense  of  progress.  As  the  road  is  smoother,  our 
sense  of  motion  is  diminished,  though  our  rate  of  travelling  is  accelerated. 
Railway  travelling,  especially  by  night,  or  in  a  tunnel,  has  familiarized 
every  one  with  this  remark.  Those  who  have  made  aeronautic  voyages 
testify  that  with  closed  eyes,  and  under  the  influence  of  a  steady  breeze 
communicating  no  oscillatory  or  revolving  motion  to  the  car,  the  sensation 
is  that  of  perfect  rest,  however  rapid  the  transfer  from  place  to  place. 

(16.)  But  it  is  on  shipboard,  where  a  great  system  is  maintained  in 
motion,  and  where  we  are  surrounded  with  a  multitude  of  objects  which 
participate  with  ourselves  and  each  other  in  the  common  progress  of  the 
whole  mass,  that  we  feel  most  satisfactorily  the  identity  of  sensation 


APPARENT  AND  REAL  MOTIONS. 


27 


between  a  state  of  motion  and  one  of  rest.  In  the  cabin  of  a  large  and 
heavy  vessel,  going  smoothly  before  the  wind  in  still  water,  or  drawn  along 
a  canal,  not  the  smallest  indication  acquaints  us  with  the  way  it  is  making. 
We  read,  sit,  walk,  and  perform  every  customary  action  as  if  we  were  on 
land.  If  we  throw  a  ball  into  the  air,  it  falls  back  into  our  hand ;  or  if 
we  drop  it,  it  alights  at  our  feet.  Insects  buzz  around  us  as  in  the  free 
air ;  and  smoke  ascends  in  the  same  manner  as  it  would  do  in  an  apart- 
ment on  shore.  If,  indeed,  we  corns  on  deck,  the  case  is,  in  some  respects, 
different ;  the  air,  not  being  carried  along  with  us,  drifts  away  smoke  and 
other  light  bodies  —  such  as  feathers  abandoned  to  it  —  apparently,  in 
the  opposite  direction  to  that  of  the  ship's  progress;  but,  in  reality,  they 
remain  at  rest,  and  we  leave  them  behind  in  the  air.  Still,  the  illusion, 
so  far  as  massive  objects  and  our  own  movements  are  concerned,  remains 
complete ;  and  when  we  look  at  the  shore,  we  then  perceive  the  effect  of 
our  own  motion  transferred,  in  a  contrary  direction,  to  external  objects — 
extemaly  that  is,  to  the  system  of  which  toe  form  a  part.        .^         ^^  ^  : . 

"  Provehimur,  portu,  terraque  urbesquo  recedunt.    "^    "'  <^'  • 

(17.)  In  order,  however,  to  conceive  the  earth  as  in  motion,  we  must 
form  to  ourselves  a  conception  of  its  shape  and  size.  Now,  an  object 
cannot  have  shape  and  size  unless  it  is  limited  on  all  sides  by  some  defi- 
nite outline,  so  as  to  admit  of  our  imagining  it,  at  least,  disconnected  from 
other  bodies,  and  existing  insulated  in  space.  The  first  rude  notion  we 
form  of  the  earth  is  that  of  a  flat  surface,  of  indefinite  extent  in  all  direc- 
tions from  the  spot  where  we  stand,  above  which  are  tlie  air  and  sky ; 
below,  to  an  indefinite  profundity,  solid  matter.  Thi^  is  a  prejudice  to  be 
got  rid  of,  like  that  of  the  earth's  immobility;  —  but  it  is  one  much 
easier  to  rid  ourselves  of,  inasmuch  as  it  originates  only  in  our  own  mental 
inactivity,  in  not  questioning  ourselves  where  we  will  place  a  limit  to  a 
thing  we  have  been  accustomed  from  infancy  to  regard  as  immensely 
large ;  and  does  not,  like  that,  originate  in  the  testimony  of  our  senses 
unduly  interpreted.  On  the  contrary,  the  direct  testimony  of  our  senses 
lies  the  other  way.  When  we  see  the  sun  set  in  the  evening  in  the  west, 
and  rise  again  in  the  east,  as  we  cannot  doubt  that  it  is  the  same  sun  we 
eee  after  a  temporary  absence,  we  must  do  violence  to  all  our  notions  of 
solid  matter,  to  suppose  it  to  have  made  its  way  through  the  substance  ot 
the  earth.  It  must,  therefore,  have  gone  under  it,  and  that  not  by  a  mere 
subterraneous  channel;  for  if  we  notice  the  points  where  it  sets  and  rises 
for  many  successive  days,  or  for  a  whole  year,  we  shall  find  them  con- 
stantly shifting,  round  a  very  large  extent  of  the  horizon ;  and,  besides, 
the  moon  and  stars  also  set  and  rise  again  in  all  points  of  the  visible 


/ 


28 


OUTLINES  Off  ASTROICOMT. 


horizon.  The  ooDclusioa  is  plain  :  the  earth  cannot  extend  indefinitely 
in  depth  downwards,  nor  iadefibitely  in  surfoee  laterally  ;  it  must  have  not 
only  bounds  in  a  horizontal  direction,  hut  also  an  under  aide  round  whioh 
the  eun,  moon,  and  stars  can  pass ;  and  that  ade  must,  at  least,  be  so  far 
like  what  we  see,  that  it  must  have  a  sky  and  sunshine,  and  a  day  when 
U  is  ni^t  to  usy  and  vice  vtrsd ;  where,  in  shorty 

— "  redU  4  noWa  Aarora,  diemq,ue  Mducft.      ^nvi'p   s»3  iuf-i-t 
Nosqiie  ubi  primus  equis  oriens  afflavit  anhelia» 
Illic  sera  rubeps  accendit  lutnioa  Vesper.  Georg. 


■'■  ^•M>&■h'^.^ 


•.',n.,) 


(18.)  As  soon  as  we  have  &miliarized  ourselves  w!t&  the  conception  of 
an  earth  without  foundations  or  fixed  supports  —  existing  insulated  in 
space  from  contact  of  every  thing  external,  it  becomes  easy  to  imagine  it 
in  motion  —  or,  rather,  difficult  to  imagine  it  otherwise :  for,  since  there 
is  nothing  to  retain  it  in  jne  place,  should  any  causes  of  motion  exist,  or 
any  forces  act  upon  it,  it  must  obey  their  impulse.  Let  us  next  see  what 
obvious  circumstances  there  are  to  help  us  to  a  knowledge  of  the  shape 
of  the  earth. 

(19.)  Let  us  first  examine  what  we  can  actually  see  of  its  shape.  Now, 
it  is  not  on  land  (unless,  indeed,  on  uncommonly  level  and  extensive 
plains),  that  we  can  see  any  thing  of  the  general  figure  of  the  earth ;  — 
the  hills,  trees,  and- other  objects  which  roughen  its  surface,  and  break 
and  elevate  the  line  of  the  horizon,  though  obviously  bearing  a  most  mi* 
nute  proportion  to  the  whole  earth,  are  yet  too  considerable  with  respect 
to  ourselves  and  to  that  small  portion  of  it  which  we  can  see  at  a  single 
view,  to  allow  of  our  forming  any  judgment  of  the  form  of  the  whole,  from 
that  of  a  part  so  disfigured.  But  with  the  surface  of  the  sea  or  any  vastly 
extended  level  plain,  the  case  is  otherwise.  If  we  sail  out  of  sight  of 
land,  whether  we  stand  on  the  deck  of  the  ship  or  climb  the  mast,  we  see 
the  surface  of  the  sea  —  not  losing  itself  in  distance  and  mist,  but  termi- 
nated by  a  sharp,  clear,  well-defined  line  or  ofjing  as  it  is  called,  which 
runs  all  round  us  in  a  circle,  having  our  station  for  its  centre.  That  this 
line  b  really  a  circle,  we  conclude,  first,  from  the  perfect  apparent  similar- 
ity of  all  its  parts ;  and,  secondly,  from  the  fact  of  all  its  parts  appearing 
at  the  same  distance  from  us,  and  that,  evidently,  a  moderate  one ;  and 
thirdly,  from  this,  that  its  apparent  diameter,  measured  with  an  instru- 
ment called  the  dip  sector,  is  the  same  (except  under  some  singular  atmos- 
pheric circumstances,  which  produce  a  temporary  dif  tortion  of  the  outline), 
in  whatever  direction  the  measure  is  taken, —  properties  which  belong  only 
to  the  circle  among  geometrical  figurfts.     If  we  ascend  a  high  eminence 


f 


THB  HORIZON  AND  ITS  DIP. 


'« 


on  a  plain  (for  instance,  one  of  tlie  Egyptian  pyramids,^  the  same  holdl 
good. 

(20.)  Masts  (tf  ships,  boweyer,  and  the  edifioed  erected  by  maa,  are. 
trifling  eminences  compared  to  what  natnre  itself  ftfibrds;  ^tna,  Tene> 
riffe,  Mowna  Roa,  are  eminences  from  which  no  contemptible  aliquot  part 
of  the  whole  earth -s  sur&oe  can  be  seen ;  but  from  these  again — in  these 
few  and  rare  occasions  when  the  transparency  of  the  air  will  permit  the 
real  boundary  of  the  horizon,  the  true  sea-line,  to  be  seen— the  very 
same  appearances  are  witnessed,  but  with  this  remarkable  addition,  viz., 
that  the  angular  diameter  of  the  visible  area,  as  measured  by  the  dip  sec- 
tor, is  materially  less  than  at  a  lower  level;  or,  in  other  words,  that  the 
apparent  size  of  the  earth  has  sensibly  diminished  as  we  have  receded 
from  its  surface,  while  yet  the  absolute  quantity  of  it  seen  at  once  has  been 
increased. 

(21.)  The  same  appearances  are  observed  universally,  in  every  part  of 
the  earth's  surface  visited  by  man.  Now,  the  figure  of  a  body  which, 
however  seen,  appears  always  circular,  can  be  no  other  than  a  sphere  or 
globe.  »-^ 

(22.)  A  diagram  will  elucidate  thb.  Suppose  the  earth  to  be  repre- 
sented by  the  sphere  LHNQ,  whose  centre  is  C,  and  let  A,  G,  M  be  sta- 
tions at  different  elevations  above  various  points  of  its  sur&oe,  represented 


\ 


80 


OUTLINBS  OF  ASTRONOMY. 


^7  "}  8i  "*  respeotively.  From  each  of  them  (as  from  M)  let  a  line  he 
drawn,  as  MNn,  a  tangent  to  the  surface  at  N,  then  will  this  line  represent 
the  visual  ray  along  which  the  spectator  at  M  will  see  the  visible  horizon ; 
and  as  this  tangent  sweeps  round  M,  and  comes  successively  into  the  posi* 
tions  MOc,  MPp,  MQ9,  the  point  of  contact  N  will  mark  out  on  the  sur- 
face  the  circle  NOPQ.  The  area  of  the  spherical  surface  comprehended 
within  this  circle  is  the  portion  of  the  earth's  surface  visible  to  a  spectator 
at  M,  and  the  angle  NMQ  included  between  the  two  extreme  visual  rays 
is  the  measure  of  its  apparent  angular  diameter.  Leaving,  at  present,  out 
of  consideration  the  effect  of  refraction  in  the  air  below  31,  of  which  more 
hereafter,  and  which  always  tends,  in  some  degree,  to  increase  that  angle, 
or  render  it  more  obtuse,  this  is  the  angle  measured  by  the  dip  sector. 
Now,  it  is  evident,  1st,  that  as  the  point  M  is  more  elevated  above  m,  the 
point  immediately  below  it  on  the  sphere,  the  visible  area,  t.  e.  the  spher- 
ical segment  or  slice  NOPQ,  increases ;  2dly,  that  the  distance  of  the  vis- 
ible horizon\.  or  boundary  of  our  view  from  the  eye,  viz.  the  line  MN, 
increases ;  and,  3dly,  that  the  angle  NMQ  becomes  Uss  obtuse,  or,  in  other 
words,  the  apparent  angular  diameter  of  the  earth  diminishes,  being  no- 
where so  great  as  180°,  or  two  right  angles,  but  falling  short  of  it  by 
some  sensible  quantity,  and  that  more  and  more  the  higher  we  ascend. 
The  figure  exhibits  three  states  or  stages  of  elevation,  with  the  horizon, 
&c.,  corresponding  to  each,  a  glance  at  which  will  explain  our  meaning; 
or,  limiting  ourselves  to  the  largei*  and  more  distinct,  MNOPQ,  let  the 
reader  imagine  nNM,  MQ^  to  be  the  two  legs  of  a  ruler  jointed  at  M,  and 
kept  extended  by  the  globe  NmQ  between  them.  It  is  clear,  that  as  the 
joint  M  is  urged  home  towards  the  surface,  the  legs  will  open,  and  the  ruler 
will  become  more  nearly  straight,  but  will  not  attain  perfect  straightness 
till  M  is  brou^^ht  fairly  up  to  contact  with  the  surface  at  m,  in  which  case 
its  whole  length  will  become  a  tangent  to  the  sphere  at  m,  as  is  the  line 

XJf. 

(23.)  This  explains  what  is  me  vnt  by  the  dip  of  the  horizon.  Mm, 
which  is  perpendicular  to  the  general  surface  of  the  sphere  at  m,  is  also 
the  direction  in  which  a  plumb-line'  would  hang;  for  it  is  an  observed 
fact,  that  in  all  situations,  in  every  part  of  the  earth,  the  direction  of  a 
plumb-liDC  is  exactly  perpsndiriular  to  the  surface  of  still  water;  and, 
moreover,  that  it  is  also  exactly  perpendicular  to  a  line  or  surface  truly 
adjusted  by  a  spirit-level'  Suppose,  then,  that  at  our  station  M  we  were 
to  adjust  a  line  (a  v/ooden  ruler  for  instance)  by  a  spirit-level,  with  perfect 
exactness;  then,  if  we  suppose  the  direction  of  this  line  indefinitely  pro- 

• '  Ofii^w,  to  terminate. 

*  See  these  instruments  described  in  Chap.  Ill, 


THB  HORIZON   AND   ITS   DIP. 


81 


longed  both  wayp,  as  XMY,  the  line  so  drawn  will  be  at  right  angles  to 
Mm,  ^nd  therefore  parallel  to  xmy,  the  tangent  to  the  sphere  at  m.  A 
spectator  placed  at  M  will  therefore  see  not  only  all  the  vault  of  the  sky 
above  this  line,  as  XZY,  but  also  that  portion  or  zone  of  it  which  lies 
between  XN  and  YQ ;  in  other  words,  his  sky  will  be  more  than  a  hemi- 
sphere by  the  zone  YQXN.  It  is  the  angular  breadth  of  this  redundant 
zone  —  the  angle  YMQ,  by  which  the  visible  horizon  appears  depressed 
below  the  direction  of  a  spirit-level — that  is  called  the  dip  of  the  horizon. 
It  is  a  correction  of  constant  use  in  nautical  astronomy. 

(24.)  From  the  foregoing  explanations  it  appears,  1st,  That  the  general 
figure  of  the  earth  (so  far  as  it  can  be  gathered  from  this  kind  of  observa- 
tion) is  that  of  a  sphere  or  glrhe.  In  this  we  also  include  that  of  the  sea, 
which,  wherever  it  extends,  covers  and  fills  in  those  inequalities  and  local 
irregularities  which  exist  on  land,  but  which  can  of  course  only  be  regarded 
as  trifling  deviations  from  the  general  outline  of  the  whole  mass,  as  we 
consider  an  orange  not  the  less  round  for  the  roughness  on  its  rind.  2dly, 
That  the  appearance  of  a  visible  horizon,  or  sea-offing,  is  a  consequence 
of  the  curvature  of  the  t  face,  and  does  not  arise  from  the  inability  of 
the  eye  to  follow  objects  to  a  greater  distance,  or  from  atmospheric  indis- 
tinctness. It  will  be  worth  while  to  pursue  the  general  notion  thus 
acquired  into  some  of  its  consequences,  by  which  its  consistency  with 
observations  of  a  different  kind,  and  on  a  larger  scale,  will  be  put  to  the 
test,  and  a  clear  conception  be  formed  of  the  manner  in  which  the  parts 
of  the  earth  are  related  to  each  other,  and  held  together  as  a  whole. 

(25.)  In  the  first  place,  then,  every  one  who  has  passed  a  little  while 
at  the  sea-side  is  aware  that  objects  may  be  seen  perfectly  well  beyond  the 
offing  or  visible  horizon — but  not  the  whole  of  them.  We  only  see  their 
upper  parts.  Their  bases  where  they  rest  on,  or  rise  out  of  the  water,  are 
hid  from  view  by  the  spherical  surface  of  the  sea,  which  protrudes  between 
them  and  ourselves.  Suppose  a  ship,  for  instance,  to  sail  directly  away 
from  our  station ;  —  at  first,  when  the  distance  of  the  ship  is  small,  a 
spectator,  S,  situated  at  some  certain  height  above  the  sea,  sees  the  whole 
of  the  ship,  even  to  the  loater  line  where  it  rests  on  the  sea,  as  at  A. 
As  it  recedes  it  diminishes,  it  is  true,  in  apparent  size,  but  still  the  whole 
is  seen  down  to  the  water  line,  till  it  reaches  the  visible  horizon  at  B. 
But  as  soon  as  it  has  passed  this  distance,  not  only  does  the  visible  por- 
tion still  continue  to  diminish  in  apparent  size,  but  the  hull  begins  to 
disappear  bodily,  as  if  sunk  below  the  surface.  When  it  has  reached  a 
certain  distance,  as  at  C,  its  hull  has  entirely  vanished,  but  the  masts 
and  sails  remain,  presenting  the  appearance  c.  But  if,  in  this  state  of 
things,  the  spectator  quickly  ascends  to  a  higher  station,  T,  whose  visible 


■'•• 


82 


OUTLINES  OF  ASTRONOMT. 


horizon  is  at  D,  tho  hull  comes  again  in  mght;  and,  when  he  dcaoends 
again  he  loses  it.    The  ship  still  receding,  the  lower  laiUi  seem  to  sink 


'^J'ii.   iVti':    i\   ■iVri'f  C  ■11'  H.lf    {A-' 

-!•■-    ,!  :■■>  '  -  .■   >   •■.  "1   r,j   .,i> 


Fig.  2. 


■Ml  M'       " 


■'  '^^^-^^n-'-'' » 


I.!' 


below  the  water,  as  at  d,  and  at  length  the  whole  disappears :  while  yet 
the  distinctness  with  which  the  last  portion  of  the  sail  d  is  seen  is  suoh 
as  to  satisfy  us  that  were  it  not  for  the  interposed  segment  of  the  Mi., 
ABCDE,  the  distance  T£  is  not  so  great  as  to  have  prevented  an  equally 
perfect  view  of  the  whole. 

(26.)  The  history  of  aeronautic  adventure  affords  a  curious  illustration 
of  the  same  principle.  The  late  Mr.  Sadler,  the  celebrated  aeronaut, 
ascended  on  one  occasion  in  a  balloon  from  Dublin,  and  was  wafted  acros<« 
the  Irish  Channel,  when,  on  his  approach  to  the  Welsh  coast,  the  balloon 
descended  nearly  to  the  surface  of  the  sea.  By  this  time  the  sun  was 
set,  and  the  shades  of  evening  began  to  close  in.  He  t'jiew  out  nearly 
all  his  ballast,  and  suddenly  sprang  upwards  to  a  gveat  height,  and  by  so 
doing  brought  his  horizon  to  dip  below  the  sun,  producing  the  whole 
phenomenon  of  a  western  sunrise.  Subsequently  descending  in  Wales, 
he  of  course  witnessed  a  second  sunset  on  the  same  evening. 

(27.)  If  we  could  measure  the  heights  and  exact  distance  of  two  sta- 
tions which  could  barely  be  discerned  from  each  other  over  the  edge  of 
the  horizon,  we  could  ascertain  the  actual  size  of  the  earth  itself:  and,  in 
fact,  were  it  not  for  the  effect  of  refraction,  by  which  we  are  enabled  to 
see  in  some  small  degree  round  the  interposed  segment  (as  will  be  here- 
after explained),  this  would  be  a  tolerably  good  method  of  ascertaining 
it.  Suppose  A  and  B  to  be  two  eminences,  whose  perpendicular  heights 
A  a  and  6  b  (which,  for  simplicity,  we  will  suppose  to  oe  exactly  equal) 
are  known,  as  well  as  their  exact  horizontal  interval  aDJ^,  by  measure- 
ment ;  then  it  is  clear  that  D,  the  visible  horizon  of  both,  will  lie  just 
half-way  between  them,  and  if  we  suppose  aDi  to  be  the  sphere  of  the 
earth,  and  C  its  centre  in  the  figure  G  D  frB,  we  know  J)b,  the  length  of 
the  arch  of  the  circle  between  D  and  h, — viz.  half  the  measured  interval, 


SIZE   OF   THE   EARTH. 


and  hB,  tho  excess  of  its  secant  above  its  radius — which  is  the  height  of 
B,  —  data  which,  by  the  solution  of  an  easy  geometrical  problem,  enable 


us  to  find  the  length  of  the  radius  D  C.  If,  as  is  really  the  ease,  we  sup- 
pose both  the  heights  and  distance  of  the  stations  inconsiderable  in  com- 
parison with  the  size  of  the  earth,  the  solution  alluded  to  is  contained  in 
the  following  proposition :  — 

The  earth's  diameter  bears  the  same  proportion  to  the  distance  of  the 
visible  horizon  from  the  eye  as  that  distance  does  to  the  height  of  the 
eye  above  the  sea  level. 

When  the  stations  are  unequal  in  height,  the  problem  is  a  little  more 
complicated. 

(28  )  Although,  as  we  have  observed,  the  effect  of  refraction  prevents 
this  from  being  an  exact  method  oi  ascertaining  the  dimensions  of  the 
earth,  yet  it  will  suffice  to  afford  such  an  approximation  to  it  as  shall  be 
of  use  in  the  present  stage  of  the  reader's  knowledge,  and  help  him  to 
many  just  conceptions,  on  which  account  we  shall  exemplify  its  applica- 
tion in  numbers.  Now,  it  appears  by  observation,  that  two  points,  each 
ten  feet  above  the  surface,  cease  to  be  visible  from  each  other  over  still 
water,  and  in  average  atmospheric  circumstances,  at  a  distance  of  about  8 
miles.  But  10  feet  is  the  528th  part  of  a  mile,  so  that  half  their  distance, 
or  4  miles,  is  to  the  height  of  each  as  4  x  528  or  2112  : 1,  and  therefore 
in  the  same  proportion  to  4  miles  is  the  length  of  the  earth's  diameter. 
It  must,  therefore,  be  equal  to  4  X  2112  =  8448,  or,  in  round  numbers, 
about  8000  miles,  which  is  not  very  far  from  the  truth. 

(29.)  Such  is  the  first  rough  result  of  an  attempt  to  ascertain  the 
earth's  magnitude ;  and  it  will  not  be  amiss,  if  we  take  advantage  of  it 
to  compare  it  with  objects  we  have  been  accustomed  to  consider  as  of  vast 
size,  so  as  to  interpose  a  few  steps  between  it  and  our  ordinary  ideas  of 
dimension.  We  havo  before  likened  the  inequalities  on  the  earth's  sur- 
face, arising  from  mountains,  valleys,  buildings,  &c.  to  the  roughnesses 
on  the  rind  of  an  orange,  compared  with  its  general  moss.  The  compari- 
3 


M  OUTLINES   OF  ASTRONOMY. 

son  i.o  qiiito  froo  from  exaggeration.  The  highest  mountain  known  hardly 
exccods  five  luilos  in  perpendicular  elevation :  this  is  only  one  ICOOth 
part  of  the  earth's  diuiucter;  consequently,  on  a  globe  of  sixteen  inches 
in  diameter,  such  a  mountuin  would  bo  represented  by  a  protuberance  of 
DO  more  than  one  hundredth  part  of  un  inch,  which  is  about  the  thickness 
of  ordinary  drawing-ptipcr.  Now,  as  there  is  no  entire  continent,  or  even 
any  very  extensive  tract  of  land,  known,  whoso  general  elevation  above 
the  sea  is  any  thing  like  half  this  quantity,  it  follows,  that  if  we  wuuid 
construct  a  correct  model  of  our  earth,  with  its  seas,  continrats,  and 
mountains,  on  a  globe  sixteen  inches  in  diameter,  the  whole  of  the  land, 
with  the  exception  of  a  few  prominent  points  and  ridges,  must  be  com- 
prised on  it  within  the  thickness  of  thin  writing-paper ;  and  the  highest 
bills  would  bo  represented  by  the  smallest  visible  grains  of  sand. 

(30.)  The  deepest  mine  existing  does  not  penetrate  half  a  mile  below 
the  purfuce :  a  scratch,  or  pin-hole,  duly  representing  it,  on  the  surface  of 
such  a  globe  as  our  model,  would  bo  imporceptiblo  without  a  magnifier. 

(ol.)  The  greatest  depth  of  sea,  probably,  does  not  very  much  exceed 
the  greatest  elevation  of  the  continents;  and  would,  of  course,  be  repre- 
sented by  an  excavation,  in  about  the  same  proportion,  into  the  substance 
of  the  globe :  so  that  the  ocean  comes  to  be  conceived  as  a  mere  film  of 
li(|uid,  such  as,  on  our  model,  would  be  left  by  a  brush  dipped  in  colour, 
and  drawn  over  those  parts  intended  to  represent  the  sea :  only,  in  so  con- 
ceiving it,  we  must  bear  in  mind  that  the  resemblance  extends  no  farther 
than  to  proportion  'n  point  of  quantity.  The  mechanical  laws  which 
would  regulate  the  distribution  and  movements  of  such  a  film,  and  its 
adhesion  to  the  surface,  are  altogether  different  from  those  which  govern 
♦''e  phenomena  of  the  sea. 

(32.)  Lastly,  the  greatest  extent  of  the  earth's  surface  which  has 
ever  been  seen  at  once  by  man,  was  that  exposed  to  the  view  of  MM. 
Biofc  and  Gay-Lussac,  in  their  celebrated  aeronautic  expedition  to  the 
enormous  height  of  25,000  feet,  or  rather  less  than  five  miles.  To  esti- 
mate the  proportion  of  the  area  visible  from  this  elevation  to  the  wholo 
earth's  surface,  we  must  have  recourse  to  the  geometry  of  the  sphere, 
which  informs  us  that  the  convex  surface  of  a  spherical  segment  is  to  the 
whole  surface  of  the  sphere  to  which  it  belongs  as  the  versed  sine  or  thick- 
ness of  the  segment  is  to  the  diameter  of  the  sphere ;  and  further,  that 
this  thickness,  in  the  case  we  are  considering,  is  almost  exactly  equal  to 
the  perpendicular  elevation  of  the  point  of  sight  above  the  surface.  The 
proportion,  therefore,  of  the  visible  area,  in  this  case,  to  the  whole  earth's 
surface,  is  that  of  five  miles  to  8000,  or  1  to  1600.  The  portion  visible 
from  -lEina,  the  Peak  of  Teneriul,  or  Mowna  Roa,  is  about  one  4000th. 


Tim    Al'MOSPlIIilUB.  it 

(83.)  When  wo  asoeini  to  uny  tnry  conaidernble  elevation  abovo  the 
Burfuoe  of  the  earth,  eithor  in  a  mIIikjh,  >r  ou  mountains,  wo  arc  niiido 
awaro,  by  many  uneasy  Honsatioiis,  of  au  in«ufficiont  supply  of  air,  Tho 
barometer,  an  instrument  wliich  informs  us  of  the  Mieight  of  air  incum- 
bent on  a  given  horizontal]  surface,  confirms  this  impression,  and  alfords 
a  direct  measure  of  the  rn  o  of  diminution  of  the  quantity  of  air  whicth  a 
given  space  includes  as  we  recede  from  tho  surface.  From  its  indicjitious 
we  learn,  that  when  we  have  ascended  to  tho  height  of  1000  foot,  wc  have 
left  below  us  about  one-thirtieth  of  tlio  whole  mass  of  the  atmosphere  :^ 
that  at  10,000  feet  of  perpendicular  elevation  (which  is  rather  less  than 
that  of  tho  summit  of  iEtna')  we  have  ascended  through  about  one-third ; 
and  at  18,000  feet  (which  is  nearly  that  of  Cotopaxi)  through  one-half 
the  material,  or,  at  least,  tho  ponderable  body  of  air  incumbent  on  tho 
earth's  surface.  From  the  progression  of  these  numbers,  as  well  as  (I 
priori,  from  tho  nature  of  the  air  itself,  which  is  compressible,  i.  c.  capa- 
ble of  being  condensed  or  crowded  into  a  smaller  space  in  proportion  to 
the  incumbent  pressure,  it  is  easy  to  see  that,  although  by  rising  still 
higher,  wo  should  continually  get  above  more  and  more  of  the  air,  and  so 
relieve  ourselves  more  and  more  from  the  pressure  with  which  it  weighs 
upon  us,  yet  the  amount  of  this  additional  relief,  or  tho  ponderable  quan- 
tity of  air  surmounted,  would  be  by  no  means  in  proportion  to  the  addi- 
tional height  ascended,  but  in  a  constantly  decreasing  ratio.  An  easy 
calculation,  however,  founded  on  our  experimental  knowledge  of  the  pro- 
perties of  air,  and  the  mechanical  laws  which  regulate  its  dilatation  and 
compression,  is  suflScient  to  show  that,  at  an  altitude  above  the  surface  of 
the  earth  not  exceeding  the  hundredth  part  of  its  diameter,  the  tenuity, 
or  rarefaction,  of  the  air  must  be  so  excessive,  that  not  only  animal  life 
could  not  subsist,  or  combustion  be  maintained  in  it,  but  that  the  most 
delicate  means  we  possess  of  ascertaining  the  existence  of  any  air  at  all 
wouM  fail  to  afford  the  slightest  perceptible  indications  of  its  presence. 

(31.)  Laying  out  of  consideration,  therefore,  at  present,  all  nice  ques- 
tions as  to  tho  probable  existence  of  a  definite  limit  to  the  atmosphere, 
beyond  which  there  is,  absolutely  and  rigorously  speaking,  no  air,  it  is 
clear,  that,  for  all  practical  purposes,  wo  may  speak  of  those  regions  which 
are  more  distant  above  tho  earth's  surface  than  the  hundredth  part  of  its 
diameter  as  void  of  air,  and  of  course  of  clouds  (which  are  nothing  but 
visible  vapours,  diflFused  and  floating  in  the  air,  sustained  by  it,  and  ren- 
dering it  iurhid  as  mud  does  water).     It  seems  probable,  from  many  indi- 

*  The  height  of  M\wi  above  tho  Mediterranean  (qs  it  results  from  a  barometrical 
measurement  of  my  own,  made  in  July,  1824,  under  very  favourable  circumstances)  ia 
10,872  English  icc\.— Author. 


36 


OUTLINES    OF   ASTRONOMY. 


cations,  that  the  greatest  height  at  \t'hieh  visible  clouds  ever  exist  does  not 
exceed  ten  miles ;  at  which  height  the  density  of  the  air  is  about  an  eighth 
part  of  what  it  is  at  the  level  of  the  sea. 

(35.)  We  are  thu^  led  to  regard  the  atmosphere  of  air,  with  the  clouds 
it  supports,  as  constituting  a  coating  of  equable  or  nearly  equable  thick- 
ness, enveloping  our  globe  on  all  sides ;  or  rather  as  an  aerial  ocean,  of 
which  the  surface  of  the  sea  and  land  constitutes  the  bed,  and  whose  infe- 
rior portions  or  strata,  within  a  few  miles  of  the  earth,  contain  by  far  the 
greater  part  of  the  whole  mass,  the  density  diminishing  with  extreme 
rapidity  as  we  recede  upwards,  till,  within  a  very  moderate  distance  (such 
as  would  be  represented  by  the  sixth  of  an  inch  on  the  model  we  have 
before  spoken  of,  and  which  is  not  more  in  proportion  to  the  globe  on 
which  it  rests,  than  the  downy  skin  of  a  peach  in  comparison  with  the 
fruit  within  it),  all  sensible  trace  of  the  existence  of  air  disappears. 

(36.)  Arguments,  however,  are  not  wanting  to  render  it,  if  not  abso- 
lutely certain,  at  least  in  the  highest  degree  probable,  that  the  surface  of 
the  aerial,  like  that  of  the  aqueous  ocean,  has  a  real  and  definite  limit,  as 
iibove  hinted  at;  beyond  which  there  is  positively  no  air,  and  above  which 
a  fresh  quantity  of  air,  could  it  be  added  from  without,  or  carried  aloft 
from  below,  instead  of  dilating  itself  indefinitely  upwards,  would,  after  a 
certain  very  enormous  but  still  finite  enlargement  of  volume,  sink  and 
merge,  as  water  poured  into  the  sea,  and  distribute  itself  among  the  mass 
beneath.  With  the  truth  of  this  conclusion,  however,  astronomy  has 
little  concern  j  all  the  effects  of  the  atmosphere  in  modifying  astronomical 
phenomena  being  the  same,  whether  it  be  supposed  of  definite  extent  or 
not. 

(37.)  Moreover,  whichever  idea  we  adopt,  within  those  limits  in  which 
it  possesses  any  appreciable  density  its  constitution  is  the  same  over  all 
points  of  the  earth's  surface ;  that  is  to  say,  on  the  great  scale,  and  leaving 
out  of  consideration  temporary  and  local  causes  of  derangement,  such  as 
winds,  and  great  fluctuations,  of  the  nature  of  waves,  which  prevail  in  it 
to  an  immense  extent.  In  other  words,  the  law  of  diminution  of  the 
air's  density  as  we  recede  upwards  from  the  level  of  the  sea  is  the  same 
in  every  column  into  which  we  may  conceive  it  divided,  or  from  whatever 
point  of  the  surface  we  may  set  out.  It  may  therefore  be  considered  as 
consisting  of  successively  superposed  strata  or  layers,  each  of  the  form  of 
a  spherical  she!',  concentric  with  the  general  surface  of  the  sea  and  land, 
and  each  of  which  is  rarer,  or  specifically  lighter,  than  that  immediately 
beneath  it;  and  ^"nser,  or  specifically  heavier,  than  that  immediately 
above  it.  This,  at  least,  is  the  kind  of  distribution  which  alone  would  be 
consistent  with  the  laws  of  the  equilibrium  of  fluids.     Inasmuch,  however^ 


it ' 


y 


THE   ATMOSPHERE. 


at 


as  the  atmosphere  is  not  in  perfect  equilibrium,  being  always  kept  in  a 
state  of  circulation,  owing  to  the  excess  of  heat  in  its  equatorial  regions 
over  that  at  the  poles,  some  slight  deviation  from  the  rigorous  expression 
of  this  law  takes  place,  and  in  peculiar  localities  there  is  reason  to  believe 
that  even  considerable  permanent  depressions  of  the  contours  of  these 
strata,  below  their  general  or  spherical  level,  subsist.  But  these  are  points 
of  consideration  rather  for  the  meteorologist  than  the  astronomer.  It 
must  be  observed,  moreover,  that  with  this  distribution  of  its  strata  the 
inequalities  of  mountains  and  valleys  have  little  concern.  These  exercise 
hardly  more  influence  in  modifying  their  general  spherical  figure  than  the 
inequalities  at  the  bottom  of  the  sea  interfere  with  the  general  sphericity 
of  its  surface.  They  would  exercise  absolutely  none  were  it  not  for  their 
effect  in  giving  another  than  horizontal  direction  to  the  currents  of  air 
constituting  winds,  as  shoals  in  the  ocean  throw  up  the  currents  which 
sweep  over  them  towards  the  surface,  and  so  in  some  small  degree  tend  to 
disturb  the  perfect  level  of  that  surface. 

(38.)  It  is  the  power  which  air  possesses,  in  common  with  all  trans- 
parent media,  of  refracting  the  rays  of  light,  or  bending  them  out  of 
their  straight  course,  which  renders  a  knowledge  of  the  constitution  of  the 
atmosphere  important  to  the  astronomer.  Owing  to  this  property,  objects 
seen  obliquely  through  it  appear  otherwise  situated  than  they  would  to  the 
same  spectator,  had  the  atmosphere  no  existence.  It  thus  produces  a 
false  impression  respecting  their  places,  which  must  be  rectified  by  ascer- 
taining the  amount  and  direction  of  the  displacement  so  apparently  pro- 
duced on  each,  before  we  can  come  at  a  knowledge  of  the  true  directions 
in  which  they  are  situated  from  us  at  any  assigned  moment. 

(39.)  Suppose  a  spectator  placed  at  A,  any  point  of  the  earth's  surface 
K  A  h;  and  let  L  Z,  M  »»,  N  n,  represent  the  successive  strata  or  layers, 
of  decreasing  density,  into  which  we  may  conceive  the  atmosphere  to  be 
divided,  and  which  are  spherical  surfaces  concentric  with  K  Ar,  the  earth's 
surface.  Let  S  represent  a  star,  or  other  heavenly  body,  beyond  the  ut- 
most limit  of  the  atmosphere.  Then,  if  the  air  were  away,  the  spectator 
would  see  it  in  the  direction  of  the  straight  line  A  S.  But,  in  reality, 
when  the  ray  of  light  S  A  reaches  the  atmosphere,  suppose  at  d,  it  will, 
by  the  laws  of  optics,  begin  to  bend  downwards,  and  fcike  a  more  inclined 
direction,  as  d  c.  This  bending  will  at  first  be  imperceptible,  owing  to 
the  extreme  tenuity  of  the  uppermost  strata;  but  as  it  advances  down- 
wards, the  strata  continually  increasing  in  density,  it  will  continually 
undergo  greater  and  greater  refraction  in  the  same  direction  j  and  thus, 
instead  of  pursuing  the  straight  line  S  d  A,  it  will  describe  a  curve  S  rf  c 
h  a,  continually  more  and  more  concave  downwards,  and  will  reach  the 


V 


OUTLINES    OF    ASTRONOMY. 


11 


earth,  not  at  A,  but  at  a  c  rtain  point  a,  nearer  to  S.  Thii  ray,  conse- 
quently, will  not  reach  the  spectator's  eye.  The  ray  by  which  he  will  see 
the  star  is,  therefore,  not  S  d  A,  but  another  ray  which,  had  there  been 
no  atmosphere,  would  have  struck  the  earth  at  K,  a  point  behind  the 
spectator ;  but  which,  being  bent  by  the  air  into  the  curve  S  D  C  B  A, 
actually  strikes  on  A.  Now,  it  is  a  law  of  optics,  that  an  object  is  seen 
in  the  direction  which  the  visual  ray  has  at  the  instant  of  arriving  at  the 
eye,  without  regiird  to  what  may  have  been  otherwise  its  course  between 
the  nhject  and  the  eye.  Hence  the  star  S  will  be  seen,  not  in  the  direc- 
tion A  S,  but  in  that  of  A  s,  a  tangent  to  the  curve  S  D  C  B  A,  at  A. 


Fig.  4. 


But  because  the  curve  described  by  the  refracted  ray  is  concave  down- 
wards, the  tangent  A  s  will  lie  above  A  S,  the  unrefracted  ray :  conse- 
quently the  object  S  will  appear  more  elevated  above  the  horizon  A  H, 
when  ween  through  the  refracting  atmosphere,  than  it  would  appear  were 
thoro  no  such  atmosphere.  Since,  however,  the  disposition  of  the  strata 
is  the  same  in  all  directions  around  A,  the  visual  ray  will  not  be  made  to 
deviate  laterally,  but  will  remain  constantly  in  the  same  vertical  plane, 
S  A  (",  passing  through  the  eye,  the  object,  and  the  earth's  centre. 

('40.)  The  eflect  of  the  air's  refraction,  then,  is  to  raise  all  the  heavenly 
bodi(>s  higher  above  the  horizon  in  appearance  than  they  are  in  reality. 
Any  i:uch  body,  situated  actually  in  the  true  horizon,  will  appear  alove  it, 
or  will  have  some  certain  apparent  altitude  (an  it  is  called).  Nay,  even 
Honio  (tf  those  actually  below  the  horizon,  and  which  would  therefore  be 
invisible  but  for  the  effect  of  refraction,  are,  by  that  effect,  raised  above  it 
and  brought  into  sight.   Thus,  the  sun,  when  situated  at  P  below  the  true 


■I  i 


BErRACTlON. 


m 


horizon,  A  H,  of  the  spectator,  becomes  visible  to  him,  as  if  it  stood  at 
p,  by  the  refracted  ray  V  q  r  t  A,  to  which  A  p  is  &  tangent. 

(41.)  The  exact  estimation  of  the  amount  of  atmospheric  refraction,  or 
the  strict  determination  of  the  angle  S  A  s,  by  which  a  celestial  object  at 
any  assigned  altitude,  H  A  S,  is  raised  in  appearance  above  its  true  place, 
is,  unfortunately,  a  very  difficult  subject  of  physical  inquiry,  and  one  on 
which  geometers  (from  whom  alone  we  can  look  for  any  information  on 
the  subject)  are  not  yet  entirely  agreed.  The  difficulty  arises  from  this, 
that  the  density  of  any  stratum  of  air  (on  which  its  refracting  power 
depends)  is  affected  not  merely  by  the  superincumbent  pressure,  but  also 
by  its  temperature  or  degree  of  heat.  Now,  although  we  know  that  as 
we  recede  from  the  earth's  surface  the  temperature  of  the  air  is  constantly 
diminishing,  yet  the  law,  or  amount  of  this  diminution  at  different  heights, 
is  not  yet  fully  ascertained.  Moreover,  the  refracting  powei  of  air  is  per- 
ceptibly affected  by  its  moisture ;  and  this,  too,  is  not  the  same  in  every 
part  of  an  aerial  column ;  neither  are  we  acquainted  with  the  laws  of  its 
distribution.  The  consequence  of  our  ignorance  on  these  points  is  to 
introduce  a  corresponding  degree  of  uncertainty  into  the  determination  of 
the  amount  of  refraction,  which  affects,  to  a  certain  appreciable  extent, 
our  knowledge  of  several  of  the  most  important  data  of  astronomy. 
The  uncertainty  thus  induced,  is,  however,  confined  within  such  very 
narrow  limits  as  to  be  no  cause  of  embarrassment,  except  in  the  most 
delicate  inquiries,  and  to  call  for  no  further  allusion  in  a  treatise  like  the 
present. 

(42.)  A  "  Table  of  Refraction,"  as  it  is  called,  or  a  statement  of  the 
amount  of  apparent  displacement  arising  from  this  cause,  at  all  altitudes, 
or  in  every  situation  of  a  heavenly  body,  from  the  horizon  to  the  zenith,^ 
or  point  of  the  sky  vertically  above  the  spectator,  and,  under  all  the 
circumstances  in  which  astronomical  observations  are  usually  performed 
which  may  influence  the  result,  is  one  of  the  most  important  and  indis- 
pensable of  ail  astronomical  tables,  since  it  is  only  by  the  use  of  such  a 
table  we  are  enabled  to  get  rid  of  an  illusion  which  must  otherwise  pervert 
all  our  notions  respecting  the  celestial  motions.  Such  have  been,  iiccorJ- 
ingly,  constructed  with  great  care,  and  arc  to  be  found  in  every  collectiou 
of  astronomical  tables.  Our  design,  in  the  present  treatise,  will  not 
admit  of  the  introduction  of  tables;  and  we  muse,  therefore,  content  our- 
selves here,  and  in  similar  cases,  with  referring  the  reader  to  works 
especially  destin»)d  to  furnish  these  useful  aids  to  calculation.     It  is,  how- 


'  From  an  Arabic  word  of  thia  signification. 
Chap.  II. 


See  this  term  technically  defined  in 


ly 


ir 


40 


OUTLINES   OF  ASTRONOMY. 


ever,  desirable  that  he  should  bear  in  mind  the  following  general  notions 
of  its  amount,  and  law  of  variations.  '    ',,'"} 

(43.)  1st.  In  the  zenith  there  is  no  refraction.  A  celestial  object, 
situated  vertically  over-head,  ia  seen  in  its  true  direction,  as  if  there  were 
no  atmosphere,  at  least  if  the  air  be  tranquil. 

2dly.  In  descending  from  the  zenith  to  the  horizon,  the  refraction  con- 
tinually increases.  Objects  near  the  horizon  appear  more  elevated  by  it 
above  thei^'  true  directions  than  those  at  a  high  altitude.    ^  ^        '  >  '• 

Sdly.  The  rate  of  its  increase  is  nearly  in  proportion  to  the  tangent  of 
the  apparent  angular  distance  of  the  object  from  the  zenith.  But  this 
rule,  which  is  not  far  from  the  truth,  at  moderate  zenith  distances,  ceases 
to  give  correct  results  in  the  vicinity  of  the  horizon,  where  the  law 
becomes  much  more  complicated  in  its  expression. 

4thly.  The  average  amount  of  refraction,  for  an  object  half-way  between 
the  zenith  and  the  horizon,  or  at  an  apparent  altitude  of  45**,  is  about  V 
(more  exactly  57")>  a  quantity  hardly  sensible  to  the  naked  eye :  out  at 
the  visible  horizon  it  amounts  to  no  less  a  quantity  than  33',  which  is 
rather  more  than  the  greatest  apparent  diameter  of  either  the  sun  or  the 
moon.  Hence  it  follows,  that  when  we  see  the  lower  edge  of  the  sun  or 
moon  just  apparently  resting  on  the  horizon,  its  whole  disk  is  in  reality 
below  it,  and  would  be  entirely  out  of  sight  and  concealed  by  the  con- 
vexity of  the  earl,h,  but  for  the  bending  round  it,  which  the  rays  of  light 
have  undergone  in  their  passage  through  the  air,  as  alluded  to  in  art.  40. 

5thly.  That  when  tbo  barometer  is  higher  than  its  average  or  mean 
state,  the  amount  of  refraction  is  greater  than  its  mean  amount ;  when 
lower,  less;  and, 

6thly.  That  in  one  and  the  same  state  of  the  barometer  the  refrac- 
tion is  greater,  the  colder  the  air.  The  variation,  owing  to  these  two 
causes,  from  its  mean  amount  (at  temp.  55",  pressure  30  inches),  are 
about  one  420th  part  of  that  amount  for  each  degree  of  the  thermometer 
of  Fahrenheit,  and  one  300th  for  each  tenth  of  an  inch  in  the  height  of 
the  barometer. 

(44.)  It  follows  from  this,  that  one  obvious  effect  of  refraction  must  bo 
to  shorten  the  duration  of  night  and  darkness,  by  actually  prolonging  the 
stay  of  the  sun  and  moon  above  the  horizon.  But  even  after  they  are  set, 
the  influence  of  the  atmosphere  still  continues  to  send  us  a  portion  of  their 
light ;  not,  indeed,  by  direct  transmission,  but  by  reflection  upon  the  vu- 
pours,  and  minute  solid  particles  which  float  in  it,  and,  perhaps,  also  on 
the  actual  material  atoms  ci  the  air  itself  To  understand  how  this  takes 
place,  we  must  recollect,  that  it  is  not  only  by  the  direct  light  of  a  lumi 
nous  object  that  we  see,  but  that  whatever  portion  of  its  light  which 


TWILIGHT. 


41/ 


3Deral  notions 


would  not  otherwise  reach  our  eyes  is  intercepted  in  its  course,  and  thrown 
back,  or  laterally,  upon  us,  becomes  to  us  a  means  of  illumination.  Such 
reflective  obstacles  always  exist  floating  in  the  air.  The  whole  course  of  a 
sun-beam  penetrating  through  the  chink  of  a  window-shutter  into  a  dark 
room  is  visible  as  a  bright  line  in  the  air;  and  even  if  it  be  stifled,  or  let 
out  through  an  opposite  crevice,  the  light  scattered  through  the  apartment 
from  this  source  is  sufficient  to  prevent  entu-e  darkness  in  the  room.  The 
luminous  lines  occasionally  seen  in  the  air,  in  a  sky  full  of  partially  bro- 
ken clouds,  which  the  vulgar  term  "  the  sun  drawing  water,"  are  simi- 
larly caused.  They  are  sunbeams,  through  apertures  in  the  clouds,  par- 
tially intercepted  and  reflected  on  the  dust  and  vapours  of  the  air  below. 
Thus  it  is  with  those  solar  rays  which,  after  the  sun  is  itself  concealed  by 
the  convexity  of  the  earth,  continue  to  traverse  the  higher  regions  of  the 
atmosphere  above  our  heads,  and  pass  through  and  out  of  it,  without  di- 
rectly striking  on  the  earth  at  all.  Some  portion  of  them  is  intr.icepted 
and  reflected  by  the  floating  particles  above  mentioned,  and  thrown  back, 
or  laterally,  so  as  to  reach  us,  and  afford  us  that  secondary  illumination, 

'        ■  Fig.  6. 


which  is  twilight.  The  course  of  s"ch  rays  will  be  immediately  understood 
from  the  annexed  figure,  in  which  A  B  C  D  is  the  earth ;  A  a  point  on  its 
surface,  where  the  sun  S  is  in  the  act  of  setting  j  its  last  lower  ray  SAM 
just  grazing  the  surface  &\.  A,  while  its  superior  rays  S  N,  S  0,  traverse 
the  atmosphere  above  A  without  striking  the  earth,  leaving  it  finally  at 
the  points  P  Q  R,  after  being  more  or  less  bent  in  passing  through  it,  the 
lower  most,  the  higher  less,  and  that  which,  like  S  R  0,  merely  grazes 
the  exterior  limit  of  the  atmosphere,  not  at  all.  Let  us  consider  several 
points,  A,  B,  C,  D,  each  more  remote  than  the  last  from  A,  and  each  mors 


42 


OUTLINES   OF  ASTRONOMY. 


deeply  involved  in  the  earth's  shadow,  which  occupies  the  whole  space 
from  A  beneath  the  lino  A  M.  Now,  A  just  receives  the  sun's  last  direct 
ray,  and,  besides,  is  illuminated  by  the  whole  reflective  atmosphere  P  Q 
K,  r.  It  therefore  receives  twilight  from  the  whole  sky.  The  point  li, 
to  which  the  sun  has  set,  receives  no  direct  solar  light,  nor  any,  direct  or 
reflected,  from  all  that  part  of  its  visible  atmosphere  which  is  below  AP  M ; 
but  from  the  lenticular  portion  P  R  z,  which  is  traversed  by  the  sun's  rays, 
and  which  lies  above  the  visible  horizon  B  R  of  B,  it  receives  a  twilight, 
which  is  strongest  at  R,  the  point  immediately  below  which  the  sun  is, 
and  fades  away  gradually  towards  P,  as  the  luminous  part  of  the  atmo- 
sphere thins  off.  At  C,  only  the  last  or  thinnest  portion,  P  Q  z  of  the 
lenticular  segment,  thus  illuminated,  lies  above  the  horizon,  0  Q,  of  that 
place ;  here,  then,  the  twilight  is  feeble,  and  confined  to  a  small  space  in 
and  near  the  horizon,  which  the  sun  has  quitted,  while  at  D  the  twilight 
has  ceased  altogether. 

(45.)  When  the  sun  is  above  the  horizon,  it  illuminates  the  atmosphere 
and  clouds,  and  these  again  disperse  and  scatter  a  portion  of  its  light  in. 
all  directions,  so  as  to  send  some  of  its  rays  to  every  exposed  point,  from 
every  point  of  the  sky.  The  generally  diffused  light,  therefore,  which  wr- 
cnjoy  in  the  daytime,  is  a  phenomenon  originating  in  the  very  same  causes 
as  the  twilight.  Were  it  not  for  the  reflective  and  scattering  power  of  the 
atmosphere,  no  objects  would  be  visible  to  us  out  of  direct  sunshine ;  every 
shadow  of  a  passing  cloud  would  be  pitchy  darkness  j  the  stars  would  be 
visible  all  day,  and  every  apartment,  into  which  the  sun  had  not  direct 
admission,  would  be  involved  in  nocturnal  obscurity.  This  scattering  action 
of  the  atmosphere  on  the  solar  light,  it  should  be  observed,  is  increased 
by  the  irregularity  of  temperature  caused  by  the  same  luminary  in  its 
different  parts,  which,  during  the  daytime,  throws  it  into  a  constant  state 
of  undulation,  and,  by  tbus  bringing  together  masses  of  air  of  very  un- 
equal temperatures,  produces  partial  reflections  and  refractions  at  their 
common  boundaries,  by  which  some  portion  of  the  light  is  turned  aside 
from  the  direct  course,  and  diverted  to  the  purposes  of  general  illumina- 
tion. 

(46.)  From  the  explanation  we  have  given,  in  arts.  39  and  40,  of  the 
nature  of  atmospheric  refraction,  and  the  mode  in  which  it  is  produced  in 
the  progress  of  a  ray  of  light  through  successive  strata,  or  layers  of  the 
atmosphere,  it  will  be  evident,  that  whenever  a  ray  passes  ohliqucli/  from 
a  higher  level  to  a  lower  one,  or  vice  versd,  its  course  is  not  rectilinear, 
but  concave  downwards ;  and  of  course  any  object  seen  by  means  of  such 
a  ray,  must  appear  deviated  from  its  true  place,  whether  that  object  be, 
like  the  celestial  bodies,  entirely  beyond  the  atmosphere,  or,  like  the  sum- 


TWILIGHT. 


43 


mits  of  mountains  seen  from  the  plains,  or  other  terrestrial  stations  at 
diflFerent  levels  seen  from  each  other,  immersed  h  it.  Every  difierence 
of  level,  accompanied,  as  it  must  be,  vrith  a  difference  of  density  in  the 
atrial  strata,  must  also  have,  corresponding  to  it,  a  certain  amount  of  re* 
fraction ;  less,  indeed,  than  what  would  be  prodi.ced  by  the  whole  atmo- 
sphere, but  still  often  of  very  appreciable,  and  even  considerable,  amount. 
This  refraction  between  terrestrial  stations  is  termed  terrestrial  re/i'mlion, 
to  distinguish  it  from  that  total  effect  which  is  only  produced  on  celestial 
objects,  or  such  as  are  beyond  the  atmosphere,  and  which  is  called  celos* 
tial  or  astronomical  refraction. 

(47.)  Another  effect  of  refraction  is  to  distort  the  visible  forms  and 
proportions  of  objects  oeen  near  the  horizon.  The  sun,  for  instance,  which 
at  a  considerable  altitude  always  appears  round,  assumes,  as  it  approuches 
the  horizon,  a  flattened  or  oval  outline ;  its  horizontal  diameter  being  vis> 
ibly  greater  than  that  in  a  vertical  direction.  When  very  near  the  horizon, 
this  flattening  is  evidently  more  considerable  on  the  lower  side  than  on  the 
upper ;  so  that  the  apparent  form  is  neither  circular  nor  elliptic,  but  a  spe- 
cies of  oval,  which  deviates  more  from  a  circle  below  than  above.  This 
singular  effect,  which  a,  one  may  notice  in  a  fine  sunset,  arises  from  the 
rapid  rate  at  which  the  refraction  increases  in  approaching  the  horizon. 
Were  every  visible  point  in  the  sun's  circumference  equally  raised  by  re- 
fraction, it  would  still  appear  circular,  tuough  displaced ;  but  the  lower 
portions  being  more  raised  than  the  upper,  the  vertical  diameter  is  thereby 
shortened,  while  the  two  extremities  of  its  horizontal  diameter  are  equally 
raised,  and  in  parallel  directions,  so  that  its  apparent  length  remains  the 
same.  The  dilated  size  (generally)  of  the  sun  or  moon,  when  seen  near 
the  horizon,  beyond  what  they  appear  to  have  when  high  up  in  the  sky, 
has  nothing  to  do  with  refraction.  It  is  an  illusion  of  the  judgment, 
arising  from  the  terrestrial  objects  interposed,  or  placed  in  close  compar- 
ison with  them.  In  that  situation  we  viev  and  judge  of  them  as  we  do 
of  terrestrial  objects — in  detail,  and  with  an  acquired  habit  of  attention 
to  parts.  Aloft  we  have  no  associations  to  guide  us,  and  their  insulation 
in  the  expanse  of  sky  leads  us  rather  to  undervalue  than  to  over-rate  their 
apparent  magnitudes.  Actual  measurement  with  a  proper  instrument  cor- 
rects our  error,  without,  however,  dispelling  our  illusion.  By  this  we 
learn,  that  the  sun,  when  just  on  the  horizon,  subtends  at  our  eyes  almost 
exactly  the  same,  and  the  moon  a  materially  less  angle,  than  when  seen 
at  a  great  altitude  in  the  sky,  owing  to  its  gi  jater  distance  from  us  in  the 
former  situation  as  compared  with  the  latter,  as  will  be  explained  farther 
on. 

(48.)  After  what  has  been  said  of  the  small  extent  Df  the  atmosphere 


44 


OUTLINES  OF  ASTRONOMY. 


in  comparison  with  the  mass  of  the  earth,  wo  shall  have  little  hesitation 
in  admitting  those  luminaries  which  people  and  adorn  the  sky,  and  which, 
while  they  obviously  form  no  part  of  the  earth,  and  receive  no  support 
from  it,  arc  yet  not  borne  along  at  random  like  clouds  upon  the  air,  nor 
drifted  by  the  winds,  to  be  external  to  our  atmosphere.  As  such  wo  have 
considered  them  while  speaking  of  their  refractions  —  as  existing  in  the 
immensity  of  space  beyond,  and  situated,  perhaps,  for  any  thing  we  can 
perceive  to  the  contrary,  at  enormous  distances  from  us  and  from  each 
other. 

(49.)  Could  a  spectator  exist  unsustained  by  the  earth,  or  any  solid 
support,  he  would  see  around  him  at  one  view  the  whole  contents  of  space 
—  the  visible  constituents  of  the  universe:  and,  in  the  absence  of  any 
means  of  judging  of  their  distances  f'om  him,  would  refer  them,  in  the 
directions  in  which  they  were  seen  from  his  station,  to  the  concave  sur- 
face of  an  imaginary  sphere,  having  his  eye  for  a  centre,  and  its  surface 
at  some  vast  indeterminate  distance.  Perhaps  he  might  judge  those 
which  appear  to  him  large  and  bright,  to  be  nearer  to  him  than  the 
smaller  and  less  brilliant;  but,  independent  of  other  means  of  judging, 
he  would  have  no  warrant  for  this  opinion,  any  more  than  for  the  idea 
that  all  were  equidistant  from  him,  and  really  arranged  on  such  a 
spherical  surface.  Nevertheless,  there  would  be  no  impropriety  in  his 
referring  their  places,  geometrically  speaking,  to  those  points  of  such  a 
purely  imaginary  sphere,  which  their  respective  visual  rays  intersect ;  and 
there  would  be  much  advantage  in  so  doing,  as  by  that  means  their  ap- 
pearance and  relative  situation  could  be  accurately  measured,  recorded, 
and  mapped  down.  The  objects  in  a  landscape  are  at  every  variety  of 
distance  from  the  eye,  yet  we  lay  them  all  down  in  a  picture  on  one 
plane,  and  at  one  distance,  in  their  actual  apparent  proportions,  and  the 
likeness  is  not  taxed  with  incorrectness,  though  a  man  in  the  foreground 
should  be  represented  larger  than  a  mountain  in  the  distance.  So  it  is 
to  a  spectator  of  the  heavenly  bodies  pictured,  projected,  or  mapped  down 
on  that  imaginary  sphere  we  call  the  sky  or  heaven.  Thus,  we  may 
easily  conceive  that  the  moon,  which  appears  to  us  as  large  as  the  sun, 
though  less  bright,  may  owe  that  apparent  equality  to  its  greater  prox- 
imity, and  may  be  really  much  less ;  while  both  the  moon  and  sun  may 
only  appear  larger  and  brighter  than  the  stars,  on  account  of  the  remote- 
ness of  the  latter. 

(50.)  A  spectator  on  the  earth's  surface  is  prevented-  by  the  great 
mass  on  which  he  stands,  from  seeing  into  all  that  portion  of  space  which 
is  below  him,  or  to  see  which  he  must  look  in  any  degree  downwards, 
It  is  true  that,  if  his  place  of  observation  be  at  a  great  elevation,  the  dip 


CHANGE  OP  HORIZON  IN  TRAVELLING. 


45 


of  the  horizon  will  bring  withii  the  scope  of  vision  a  little  more  thai:  a 
hemisphere,  and  refraction,  wherever  he  may  be  situated,  will  enable  him 
to  look,  aa  it  were,  a  little  round  the  corner ;  but  the  zone  thus  added  to 
his  visual  range  can  hardly  over,  unless  *-  very  extraordinary  circura- 
stances,  exceed  a  couple  of  degrees  in  breadth,  and  is  always  ill  seen  on 
account  of  the  vapours  near  the  horizon.  Unless,  then,  by  a  change  of 
his  geographical  situation,  he  should  shift  his  horizon  (which  is  always  a 
plane  passing  through  his  eye,  and  touching  thQ  spherical  convexity  of 
the  earth) ;  or  unless,  by  some  movements  proper  to  the  heavenly  bodies, 
they  should  of  themselves  come  above  his  horizon ;  or,  lastly,  unless,  by 
somo  rotation  of  the  earth  itself  on  its  centre,  the  point  of  its  surface 
which  he  occupies  should  be  carried  round,  and  presented  towards  a  dif- 
ferent region  of  space ;  he  would  never  obtain  a  sight  of  almost  one  half 
the  objects  external  to  our  atmosphere.  But  if  any  of  these  cases  be 
supposed,  more,  or  all,  may  come  into  view  according  to  the  circum- 
stances. 

(51.)  A  t'aveller,  for  example,  shifting  his  locality  on  our  globe,  will 
obtain  a  view  of  celestial  objects  invisible  from  his  original  station,  in  a 
way  which  may  be  not  inaptly  illustrated  by  comparing  him  to  a  person 
standing  in  a  park  close  to  a  large  tree.  The  massive  obstacle  presented 
by  its  trunk  cuts  off  his  view  of  all  those  parts  of  the  landscape  which  it 

"    Fig.  6.     ;     . 


us,  we  may 


n     m 


occupies  as  an  object  j  but  by  walking  round  it  a  complete  successive  view 
of  the  whole  panorama  may  be  obtained.  Just  in  the  same  way,  if  we 
set  off  from  any  station,  as  London,  and  travel  southwards,  we  shall  not 
fail  to  notice  that  many  celestial  objects  which  are  never  seen  from 
London  come  successively  into  view,  as  if  rising  up  above  the  horizon, 
night  after  night,  from  the  south,  although  it  is  in  reality  our  horizon. 


16 


OUTLINES   OP   ASTRONOMY. 


which,  travolliag  with  us  southwards  round  the  sphere,  siuks  in  succes- 
sion beneath  them.  The  novelty  and  splendour  of  fresh  constellations 
thus  gradually  brought  into  view  in  the  clear  calm  nights  of  tropical 
climates,  in  long  voyages  to  the  south,  is  dwelt  upon  by  all  who  have 
enjoyed  this  spectacle,  and  never  fails  to  impress  itself  on  the  recollection 
among  the  most  delightful  and  interesting  of  the  associations  connected 
with  extensive  travel.  A  glance  at  the  accompanying  figure,  exhibiting 
three  successive  stations  of  a  traveller,  A,  B,  0,  with  the  horizon  corre- 
sponding to  each,  will  place  this  process  in  clearer  evidence  than  any 
description. 

(52.)  Again :  suppose  the  earth  itself  to  have  a  motion  of  rotation  on 
its  centre.  It  is  evident  that  a  spectator  at  rest  (as  it  appears  to  him) 
on  any  part  of  it  will,  unperceived  by  himself,  be  carried  round  with  it : 
unperceived,  we  say,  because  his  horizon  will  constantly  contain,  and  bo 
limited  by,  the  same  terrestrial  objects.  He  will  have  the  same  land- 
scape constantly  before  his  eyes,  in  which  all  the  familiar  objects  in  it, 
which  serve  him  for  landmarks  and  directions,  retain,  with  respect  to 
himself  or  to  each  other,  the  same  invariable  situations.  The  perfect 
smoothness  and  equality  of  the  motion  of  so  vast  a  mass,  in  which  every 
object  ho  sees  around  him  participates  alike,  will  (art.  15)  prevent  his 
CDtcrtainiag  any  suspicion  of  his  actual  change  of  place.  Yet,  with 
respect  to  external  objects,  —  that  is  to  say,  all  celestial  ones  which  do 
not  participate  it  the  supposed  rotation  of  the  earth, — his  horizon  will 
have  been  all  the  while  shifting  in  its  relation  to  them,  precisely  as  in 
tlie  case  of  our  traveller  in  the  foregoing  article.  Rficurring  to  the  figure 
of  that  article,  it  is  evidently  the  same  thing,  so  far  as  their  visibility  is 
concerned,  whether  he  has  been  carried  by  the  earth's  rotation  succes- 
sively into  the  situations  A,  B,  C ;  or  whether,  the  earth  remaining  at 
rest,  he  has  transferred  himself  personally  along  its  surface  to  those 
stations.  Our  spectator  in  the  park  will  obtain  precisely  the  same  view 
of  the  landscape,  whether  he  walk  round  the  tree,  or  whether  we  suppose 
it  sawed  oflF,  and  made  to  turn  on  an  upright  pivot,  while  he  stands  on  a 
projecting  step  attached  to  it,  and  allows  himself  to  be  carried  round  by 
its  motion.  The  only  difference  will  be  in  his  view  of  the  tree  itself,  of 
which,  in  the  former  case,  he  will  see  every  part,  but,  in  the  latter,  only 
that  portion  of  it  which  remains  constantly  opposite  to  him,  and  imme- 
diately under  his  eye. 

(53.)  By  such  a  rotation  of  the  earth,  then,  as  we  have  supposed,  the 
horizon  of  a  stationary  spectator  will  be  constantly  depressing  itself  below 
those  objects  which  lie  in  that  region  of  spaee  towards  which  the  rotation 
is  carrying  him,  and  elevating  itself  above  those  in  the  opposite  quarter, 


DIURNAL   ROTATION   OP  TUB   EARTH. 


47 


admitting  into  view  the  former,  and  successively  hiding  the  latter.  As 
the  horizon  of  every  such  spectator,  however,  appears  to  him  motionless, 
all  such  cuunges  will  be  referred  by  him  to  a  motion  in  the  objects  them- 
selves so  successively  disclosed  and  concealed.  In  place  of  his  horizon 
approaching  the  stars,  therefore,  ho  will  judge  the  stars  to  approach  his 
horizon ;  and  when  it  passes  over  and  hides  any  of  them,  ho  will  consider 
tbcui  as  having  sunk  below  it,  or  set;  while  those  it  has  just  disclosed, 
and  from  which  it  is  receding,  will  seem  to  be  rising  above  it. 

(54.)  If  wo  suppose  this  rotation  of  the  earth  to  continue  in  one  and 
the  same  direction,  —  that  is  to  say,  to  be  performed  round  one  and  the 
same  axis,  till  it  has  completed  an  entire  revolution,  and  come  back  to  the 
position  from  which  it  set  out  when  the  spectator  be^an  his  observations, 
—  it  is  manifest  that  every  thing  will  then  bo  in  precisely  the  same  rela- 
tive position  as  at  the  outset :  all  the  heavenly  bodies  will  appear  to  occupy 
the  same  places  in  the  concave  of  the  sky  which  they  did  at  that  instant, 
except  such  as  may  have  actually  moved  in  the  interim ;  and  if  the  rota- 
tion still  continue,  the  same  phenomena  of  their  successive  rising  and 
setting,  and  return  to  the  same  places,  will  continue  to  be  repeated  in  the 
same  order,  and  (if  the  velocity  of  rotation  be  uniform)  in  equal  intervals 
of  time,  ad  infinium. 

(55.)  Now,  in  this  we  have  a  lively  picture  of  that  grand  phenomenon, 
the  most  important  beyond  all  comparison  which  nature  presents,  the  daily 
rising  and  setting  of  the  sun  and  stars,  their  progress  through  the  vault 
of  the  heavens,  and  their  return  to  the  same  apparent  places  ait  the  same 
hours  of  the  day  and  night.  The  accomplishment  of  this  restoration  in 
tlie  regular  interval  of  twenty-four  hours  is  the  first  instance  we  encounter 
of  that  great  law  of  periodicity,^  which,  as  we  shall  see,  pervades  all 
astronomy;  by  which  expression  we  understand  the  continual  reproduction 
of  the  same  phenomena,  in  the  same  order,  at  equal  intervals  of  time. 

(56.)  A  free  rotation  of  the  earth  round  its  centre,  if  it  exist  and  bo 
performed  in  consonance  with  the  same  mechanical  laws  which  obtain  in 
the  motions  of  masses  of  matter  under  our  immediate  control,  and  within 
our  ordinary  experience,  must  be  such  as  to  satisfy  two  essential  conditions. 
It  must  be  invariable  in  its  direction  with  respect  to  the  sphere  itself,  and 
uniform  in  its  velocity.  The  rotation  must  be  performed  round  an  axis 
or  diameter  of  the  sphere,  whose  poles  or  extremities,  where  it  meets  the 
surface,  correspond  always  to  the  same  points  on  the  sphere.  Modes  of 
rotation  of  a  solid  body  under  the  influence  of  external  agency  are  con- 
ceivable, in  which  the  poles  of  the  Imaginary  line  or  axis  about  which  it 

'  Wtfloim,  a  going  round,  a  circulation  ov  revolution. 


48 


OUTLINES   OF  ASTRONOMY. 


is  at  ai.y  municnt  revolving  shall  hold  no  fizod  places  on  the  surfaco,  but 
shift  upon  it  every  moment.  Such  changes,  however,  are  inconsistent 
mth.  the  idea  of  u  rotation  of  a  body  of  regular  figure  about  its  axis  of 
symmetry,  performed  in  free  space,  and  without  resistance  or  obstruction 
from  any  suiTounding  medium,  or  disturbing  influences.  The  complete 
absence  of  such  obstructions  draws  with  it,  of  necessity,  the  strict  fulfil* 
mcnt  of  the  two  conditions  above  mentioned. 

(57.)  Now,  these  conditions  are  in  perfect  accordance  with  what  we 
observe,  and  what  recorded  observation  teachej  us,  in  respect  of  the  diur- 
nal motions  of  the  heavenly  bodies.  We  have  no  reason  to  believe,  from 
history,  that  any  sensible  change  has  taken  place  since  the  earliest  ages  in 
the  interval  of  time  elapsing  between  two  successive  returns  of  the  same 
star  to  the  same  point  of  the  sky ;  or,  rather,  it  is  demonstrable  from 
astronomical  records  that  no  such  change  has  taken  place.  And  with 
respect  to  the  other  condition, —  the  permanence  of  the  axis  of  rotation, 

—  the  appearances  which  any  alteration  in  that  respect  must  produce, 
would  bo  marked,  as  we  shall  presently  show,  by  a  corresponding  change 
of  a  very  obvious  kind  in  the  apparent  motions  of  the  stars ;  which,  again, 
history  decidedly  declares  them  not  to  have  undergone. 

(58.)  But,  before  wo  procued  to  examine  more  in  detail  how  the  hypo- 
thesis of  the  rotation  of  the  earth  about  an  axis  accords  with  the  phe- 
nomena which  the  diurnal  motion  of  the  heavenly  bodies  offers  to  our 
notice,  ic  will  be  proper  to  describe,  with  precision,  in  what  that  diurnal 
motion  consists,  and  how  far  it  is  participated  in  by  them  all ;  or  whether 
any  of  them  form  exceptions,  wholly  or  partially,  to  the  common  analogy 
of  the  rest.  We  will,  therefore,  suppose  the  reader  to  station  himself,  on 
a  clear  evening,  just  after  sunset,  when  the  first  stars  begin  to  appear,  in 
some  open  situation  whence  a  good  general  view  of  the  heavens  can  be 
obtained.  He  will  then  perceive,  above  and  around  him,  as  it  were,  a 
vast  concave  hemispherical  vault,  beset  with  stars  of  various  magnitudes, 
of  which  the  brightest  only  will  first  c^trh  his  attenti'^n  in  the  twilight; 
and  more  and  more  will  appear  i^s  tho  liarkness  increnses,  till  the  whole 
sky  is  over-spangled  with  them,  Wlii«?u  he  has  awhile  admired  the  calm 
magnificence  of  this  glorious  spx^ctaw-le,  tiie  theme  of  so  much  song,  and 
of  so  much  thought, — a  spectacU^  which  no  one  can  view  without  emotion, 
and  without  a  longing  desire  to  know  something  of  its  nature  and  purport, 

—  let  him  fix  his  attention  more  particularly  na  a  few  of  the  most  bril- 
liant stars,  such  a^  he  cannoe  fail  to  recognize  .^ain  without  mistake  after 
looking  away  from  them  for  some  time,  and  Wt  him  refer  their  apparent 
situations  to  sonic  sui'  mnding  objects,  as  butudings,  trees,  &c.,  selecting 
purposely  such  as  are  in  different  quarters  of  hin  horizon.     On  comparing 


APPARENT   DIURNAL   MOTION. 


49 


)  appear,  m 


them  again  with  their  respective  points  of  reference,  after  a  moderate 
interval,  as  the  night  advances,  ho  will  not  fail  to  perceive  that  they  have 
changed  their  places,  and  advanced,  as  by  a  general  movement,  in  a  west- 
ward direction ;  those  towards  the  eastern  quarter  appearing  to  rise  or 
recede  from  the  horizon,  while  those  which  lie  towards  the  west  will  be 
seen  to  approach  it  j  and,  if  watched  long  enough,  will,  for  the  most  part, 
finally  sink  beneath  it,  and  disappear ;  while  others,  in  the  eastern  quarter, 
will  be  seen  to  rise  as  if  out  of  the  earth,  and,  joining  in  the  general 
procession,  will  take  their  course  with  the  rest  tov.ards  the  opposite 
quarter. 

(59.)  If  he  persist  for  a  consideiable  time  in  watching  their  motions, 
on  the  same  or  on  several  successive  nights,  he  will  perceive  that  each  star 
appears  to  describe,  as  far  as  its  course  lies  above  the  horizon,  a  circle  in 
the  sky ;  that  the  circles  so  described  are  not  of  the  same  magnitude  for 
all  the  stars ;  and  that  those  described  by  different  stars  differ  greatly  in 
respect  of  the  parts  of  them  which  lie  above  the  horizon.  Some,  which 
lie  towards  the  quarter  of  the  horizon  which  is  denominated  the  South,' 
only  remain  for  a  short  time  above  it,  and  disappear,  after  describing  in 
sight  only  the  small  upper  segment  of  their  diurnal  circle ;  others,  which 
rise  between  the  south  and  east,  describe  larger  segments  of  their  circles 
above  the  horizon,  remain  proportionally  longer  in  sight,  and  set  precisely 
as  far  to  the  westward  of  south  as  they  rose  to  the  eastward ;  while  such 
as  rise  exactly  in  the  east  remain  just  twelve  hours  visible,  describe  a 
semicircle,  and  set  exactly  in  the  west.  With  those,  again,  which  rise 
between  the  east  and  north,  the  same  law  obtains;  at  least,  as  far  as 
regards  the  time  of  their  remaimug  above  the  horizon,  and  the  proportion 
of  the  vinible  segment  of  \Mt»v  diurnal  circles  to  their  whole  oircum- 
ferencos.  Both  go  on  inor««siD^ ;  they  remain  in  view  more  than  twelve 
hours,  and  their  visible  diomai  arcs  are  more  than  semicircles.  But  the 
magnitudes  of  the  circ^  themselves  diminish,  as  we  go  from  the  cast, 
northward ;  the  greatest  of  all  the  circles  being  described  by  those  which 
rise  exactly  in  the  east  point.  Carrying  his  eye  farther  northwards,  he 
will  notice,  at  length,  stars  which,  in  their  diurnal  motion,  just  graze  the 
horizon  at  its  north  point,  or  only  dip  below  it  for  a  moment ;  while  others 
never  reach  it  at  all,  but  continue  always  above  it,  revolving  in  entire 
circles  round  one  point  called  the  pole,  which  appears  to  be  the  common 
centre  of  all  their  motions,  and  which  alone,  in  the  whole  heavens,  may 
be  considered  immoveabae.  Not  that  this  point  is  marked  by  any  star 
It  is  a  purely  imaginary  centre ;  but  there  is  near  it  one  considerably 

'  We  suppose  our  observer  to  be  stationed  in  some  northern  latitude ;  some  where 
in  Europe,  for  example. 

4 


;/ 


S^ 


OUTLINES   OP  ASTRONOMY. 


bright  star,  called  the  Pole  Star,  which  is  easily  recognized  by  the  very 
small  circle  it  describes;  so  small,  indeed,  that,  without  paying  particular 
attention,  and  referring  its  position  very  nicely  to  some  fixed  mark,  it  may 
easily  be  supposed  at  rest,  and  be,  itself,  mistaken  for  the  common  centre 
about  which  all  the  others  in  that  region  describe  their  circles ;  or  it  may 
be  known  by  its  configuration  with  a  very  splendid  and  remarkable  con- 
stellation or  group  of  stars,  called  by  astronomers  the  Great  Bear. 

(60.)  He  will  further  observe,  that  the  apparent  relative  situations  of 
all  the  stars  among  one  another,  is  not  changed  by  their  diurnal  motion. 
In  whatever  parts  of  their  circles  they  are  observed,  or  at  whatever  hour 
of  the  night,  they  form  with  each  other  the  same  identical  groups  or  con- 
figur°,tions,  to  which  the  name  of  constellations  has  been  given.  It 
is  true,  that,  in  different  parts  of  their  course,  these  groups  stand  dif- 
ferently witli  respect  to  the  horizon ;  and  those  towards  the  north,  when 
in  the  course  of  their  diurnal  movement  they  pass  alternately  above  and 
below  that  common  centre  of  motion  described  in  the  last  article,  becoms 
actually  inverted  with  respect  to  the  horizon,  while,  on  the  other  hand, 
they  always  turn  the  same  points  towards  the  pole.  In  short,  he  will 
perceive  that  the  whole  assemblage  of  stais  visible  at  once,  or  in  succes- 
sion, in  the  heavens,  may  be  regarded  as  one  great  constellation,  whicli 
seems  to  revolve  with  a  uniform  motion,  as  if  it  formed  one  coherent 
mass ;  or  as  if  it  were  attached  to  the  internal  surface  of  a  vast  hollow 
sphere,  having  the  earth,  or  rather  the  spectator,  in  its  centre,  and  turning 
round  an  axis  inclined  to  his  horizon,  so  as  to  pass  through  that  fixed 
point  or  pole  already  mentioned. 

(61.)  Lastly,  he  will  notice,  if  he  have  patience  to  outwatch  a  long 
winter's  night,  commencing  at  the  earliest  moment  when  the  stars  appear, 
and  continuing  till  morning  twilight,  that  those  stars  which  ho  observed 
setting  in  the  west  have  again  risen  in  tho  east,  while  those  which  were 
rising  when  he  first  began  to  notice  them  have  completed  their  course,  and 
are  now  set ;  and  that  thus  the  hemisphere,  or  a  great  part  of  it,  which 
was  then  above,  is  now  beneath  him,  and  its  place  supplied  by  that  which 
was  at  first  under  his  feet,  which  he  will  thus  discover  to  be  no  less 
copiously  furnished  with  stars  than  the  other,  and  bespangled  with  groups 
no  less  permanent  and  distinctly  recognizable.  Thus  he  will  learn  that 
the  great  constellation  that  we  have  above  spoken  of  as  revolving  round 
the  pole  is  co-extensive  with  the  whole  surface  of  the  sphere,  being  in 
reality  nothing  less  than  a  universe  of  luminaries  surrounding  the  earth 
on  all  sidest  and  brought  in  succession  before  his  view,  and  referred  (each 
luminary  according  to  its  own  visual  ray  or  direction  from  his  eye)  to  the 
imaginary  spherical  surface,  of  which  he  himself  occupies  the  centre. 


^.-■'J^'!^'.  ■■'-:'*''-' ;f.',^'i'r''fS'-?^'7i^'^^'l^1f'^-?-' 


' ^-'-r*'!" 


APPARENT  DIURNAL   MOTION. 


51 


(See  art.  49.)  There  is  always,  therefore  (he  would  justly  argue),  a  star- 
bespangled  canopy  over  his  head,  by  day  as  well  as  by  night,  only  that 
the  glare  of  daylight  (which  he  perceives  gradually  to  efface  the  stars  as 
the  morning  twilight  comes  on)  prevents  them  from  being  seen^  Aad 
such  is  really  the  case.  The  stare  actually  continue  visible  through  teles- 
copes in  the  day-time ;  and,  in  proportion  to  the  power  of  the  instrument, 
not  only  the  largest  and  brightest  of  them,  but  even  those  of  inferior 
lustre,  such  as  scarcely  strike  the  eye  at  night  as  at  all  conspicuous,  are 
readily  found  and  followed  even  at  noonday, — unless  in  that  part  of  the 
sky  which  is  very  near  the  sun, —  by  those  who  possess  the  means  of 
pointing  a  telescope  accurately  to  the  proper  places.  Indeed,  from  the 
bottoms  of  deep  narrow  pits,  such  as  a  well,  or  the  shaft  of  a  mine,  such 
bright  stars  as  pass  the  zenith  may  even  be  discerned  by  the  naked  eye ; 
and  we  have  ourselves  hoard  it  stated  by  a  celebrated  optician,  that  the 
earliest  circumstance  which  drew  his  attention  to  astronomy  was  the 
regular  appearance,  at  a  certain  hour,  for  several  successive  days,  of  a 
considerable  star,  through  the  shaft  of  a  chimney.  Venus  in  our  climate, 
and  even  Jupiter  in  the  clearer  skies  of  tropical  countries,  are  often 
visible,  without  any  artificial  aid,  to  the  naked  eye  of  one  who  knows 
nearly  where  to  look  for  them.  During  total  eclipses  of  the  sun,  the 
larger  stars  also  appear  in  their  proper  situations. 

(62.)  But  to  return  to  our  incipient  astronomer,  whom  we  left  contem- 
plating the  sphere  of  the  heavens,  as  completed  in  imagination  beneath 
his  feet,  and  as  rising  up  from  thence  in  its  diurnal  course.  There  is  one 
portion  or  segment  of  this  sphere  of  which  he  will  not  thus  obtain  a  view. 
As  there  is  a  segment  towards  the  north,  adjacent  to  the  pole  above  his 
horizon,  in  which  the  stare  never  set,  so  there  is  a  corresponding  segment, 
about  which  the  smaller  circles  of  the  more  southern  stare  are  described, 
in  which  they  never  rise.  The  stare  which  border  upon  the  extreme 
circumference  of  this  segment  just  graze  the  southern  point  of  his  hori- 
zon, and  show  themselves  for  a  few  moments  above  it,  precisely  as  those 
near  the  circumference  of  the  northern  segment  graze  his  northern  hori- 
zon, and  dip  for  a  moment  below  it,  to  re-appear  immediately.  Every 
point  in  a  spherical  surface  has,  of  couree,  another  diametrically  opposite 
to  it ;  and  as  the  spectator's  horizon  divides  his  sphere  into  two  hemi- 
spheres—  a  superior  and  inferior  —  there  must  of  necessity  exist  a  de- 
pressed pole  to  the  south,  corresponding  to  the  elevated  one  to  the  north, 
and  a  portion  surrounding  it,  perpetually  beneath,  as  there  is  anotber 
surrounding  the  north  pole,  perpetually  above  it. 

"  Hie  vertex  nobis  semper  sublimis ;  at  ilium 
Siib  pedibuB  nox  atra  videt,  manesque  profundi."— Viroil. 


52 


OUTLINES  OP  ASTRONOMY. 

One  pole  rides  high,  one,  plunged  beneath  the  main, 
Seeks  the  deep  night,  and  Pluto's  dusky  reign. 


\ 


(63.)  To  get  sight  of  thb  segment,  he  must  travel  southwards.  In  so 
doing,  a  new  set  of  phenomena  come  forward.  In  proportion  as  he 
advances  to  the  south,  some  of  those  constellations  which,  at  his  original 
station,  barely  grazed  the  northern  horizon,  will  be  observed  to  sink  below 
it  and  set ;  at  first  remaining  hid  only  for  a  very  short  time,  but  gradually 
for  a  longer  part  of  the  twenty-four  hours.  They  will  continue,  however, 
to  circulate  about  the  same  point  —  that  is,  holding  the  same  invariable 
position  vjiih  respect  to  them  in  the  concave  of  the  heavens  among  the 
stars ;  but  this  point  itself  will  become  gradually  depressed  with  respect 
to  the  spectator's  horizon.  The  axis,  in  short,  about  which  the  diurnal 
motion  is  performed,  will  appear  to  have  become  continually  less  and  less 
inclined  to  the  horizon ;  and  by  the  same  degrees  as  the  northern  pole  is 
depressed  the  southern  will  rise,  and  constellations  surrounding  it  will 
come  into  view ;  at  first  momentarily,  but  by  degrees  for  longer  and  longer 
times  in  each  diurnal  revolution  —  realizing,  in  short,  what  we  have 
already  stated  in  art.  51. 

(64.)  If  he  travel  continually  southwards,  he  will  at  length  reach  a 
line  on  the  earth's  surface,  called  the  eqtmtor,  at  any  point  of  which, 
indifierently,  if  he  take  up  his  station  and  recommence  his  observations, 
he  will  find  that  he  has  both  the  centres  of  diurnal  motion  in  his  horizon, 
occupying  opposite  points,  the  northern  pole  having  been  depressed,  and 
the  southern  raised ;  so  that,  in  this  geographical  position,  the  diurnal 
rotation  of  the  heavens  will  appear  to  him  to  be  performed  about  a  hori< 
zontal  axis,  every  star  describing  half  its  diurnal  circle  above  and  half 
beneath  his  horizon,  remaining  alternately  visible  for  twelve  hours,  and 
concealed  during  the  same  interval.  In  this  situation,  no  part  of  the 
heavens  is  concealed  from  his  successive  view.  In  a  night  of  twelve  hours 
(supposing  such  a  continuance  of  darkness  possible  at  the  equator)  the 
whole  sphere  will  have  passed  in  review  over  him  —  the  whole  hemisphere 
with  which  he  began  his  night's  observation  will  have  been  carried  down 
beneath  him,  and  the  entire  opposite  one  brought  up  from  below. 

(65.)  If  he  pass  the  equator,  and  travel  still  farther  southwards,  the 
southern  poles  of  the  heavens  will  become  elevated  above  his  horizon,  and 
the  northern  will  sink  below  it ;  and  the  more,  the  farther  he  advances 
southwards;  and  when  arrived  at  a  station  as  far  to  the  south  of  the 
equator  as  that  from  which  he  started  was  to  the  north,  he  will  find  the 
whole  phenomena  of  the  heavens  reversed.  The  stars  which  at  his  origi- 
nal station  described  their  whole  diurnal  circles  above  his  horizon,  and 
never  set,  now  describe  them  entirely  below  it,  and  never  rise,  but  remain 


PARALLACTIC   MOTION. 


SB 


constantly  invisible  to  him ;  and  vice  versd,  those  stars  which  at  his  former 
station  he  never  saw,  he  will  now  never  cease  to  see. 

(66.)  Finally,  if,  instead  of  advancing  southwards  from  his  first  station, 
he  travel  northwards,  he  will  observe  the  northern  pole  of  the  heavens  to 
become  more  elevated  above  his  horizon,  and  the  southern  more  depressed 
below  it.  In  consequence,  his  hemisphere  will  present  a  less  variety  of 
stars,  because  a  greater  proportion  of  the  whole  surface  of  the  heavens 
remains  constantly  visible  or  constantly  invisible :  the  circle  described  by 
each  star,  too,  becomes  more  ueorly  parallel  to  the  horizon ;  and,  in  short, 
every  appearance  leads  to  suppose  that  could  he  travel  far  enough  to  the 
north,  he  would  at  length  attain  a  point  vertically  under  the  northern  pole 
of  the  heavens,  at  which  none  of  the  stars  would  either  rise  or  set,  but 
each  would  circulatiC  round  the  horizon  in  circles  parallel  to  it.  Many 
endeavours  have  been  made  to  reach  this  point,  which  is  called  the  north 
pole  of  the  earth,  but  hitherto  without  success ;  a  barrier  of  almost  insur- 
mountable difficulty  being  presented  by  the  increasing  rigour  of  the 
climate  :  but  a  very  near  approach  to  it  has  been  made ;  and  the  pheno- 
mena of  thu:'  .  "^  ons,  though  not  precisely  such  as  we  have  described  as 
what  must  g  --  it  the  pole  itself,  have  proved  to  be  in  exact  correspon- 
dence with  its  near  proximity.  A  similar  remark  applies  to  the  south 
pole  of  the  earth,  which,  however,  is  more  unapproachable,  or,  at  least, 
has  been  less  nearly  approached,  than  the  north. 

(67.)  The  above  is  an  account  of  the  phenomena  of  the  diurnal  motion 
of  the  stars,  as  modified  by  difierent  geographical  situations,  not  grounded 
on  any  speculation,  but  actually  observed  and  recorded  by  travellers  and 
voyagers.  It  is,  however,  in  complete  accordance  with  the  hypothesis  of 
a  rotation  of  the  earth  round  a  fixed  axis.  In  order  to  show  this,  how- 
ever, it  will  be  necessary  to  premise  a  few  observations  on  parallactic 
motion  in  general,  and  on  the  appearances  presented  by  an  assemblage  of 
remote  objects,  when  viewed  from  different  parts  of  a  small  and  circum- 
scribed station. 

(68.)  It  has  been  shown  (art.  16)  that  a  spectator  in  smooth  motion, 
and  surrounded  by,  and  forming  part  of,  a  great  system  partaking  of 
the  same  motion,  is  unconscious  of  his  own  movement,  and  transfers  it  in 
idea  to  objects  external  and  unconnected,  in  a  contrary  direction;  those 
which  he  leaves  behind  appearing  to  recede  from,  and  those  which  he 
advances  towards  to  approach,  him.  Not  only,  however,  do  external 
objects  at  rest  appear  in  motion  generally,  with  respect  to  ourselves  when 
we  are  in  motion  among  them,  but  they  appear  to  move  one  amcag  the 
other  —  they  shift  their  relative  apparent  places.  Let  any  one  travelling 
rapidly  along  a  high  road  fix  his  eye  steadily  on  any  object,  but  at  the 


54 


OUTLINES  OF  ASTRONOMY. 


same  time  not  entirely  withdraw  hia  attention  from  the  general  landscape, 
— he  will  see,  or  think  he  sees,  the  whole  landscape  thrown  into  rotation^ 
and  moving  round  that  object  as  a  centre;  all  objects  between  it  and 
himself  appearing  to  move  backwards,  or  the  contrary  way  to  his  own 
motion;  and  all  beyond  it,  forwards,  or  in  the  direction  in  which  he 
jQOves:  but  let  ^  'i  withdraw  his  eye  from  that  object,  and  fix  it  on 
another,  —  a  nea  ur  one,  for  instance,  —  immediately  the  appearance  of 
rotation  shifts  also,  and  the  apparent  centre  about  which  this  illusive 
circulation  is  peiformed  is  transferred  to  the  new  object,  which,  for  the 
moment,  appears  to  rest.  This  apparent  change  of  situation  of  objects 
with  respect  to  one  another,  arising  from  a  motion  of  the  spectator,  is 
called  a  parall<iAdic  motion.  Tc  see  the  reason  of  it  we  must  consider 
that  the  position  of  every  object  is  referred  by  us  to  the  surface  of  an 
imaginary  sphere  oi  an  indefinite  radius,  having  our  eye  for  its  centre : 


\   J 


Fig.  7. 


<ind,  as  we  advance  in  any  direction,  A  B,  carrying  this  imaginary  sphere 
along  with  us,  the  visual  rays  A  P,  A  Q,  by  which  objects  are  referred  to 
its  surface  (at  C,  for  instance),  shift  their  positions  with  respect  to  the 
line  ia  which  we  move,  A  B,  which  serves  as  an  axis  or  line  of  reference, 
and  assume  new  positions,  B  P  j),  B  Q  g-,  revolving  round  their  respective 
objects  as  centres.  Their  intersections,  therefore,  p,  q,  with  our  visual 
sphere,  will  appear  to  recede  on  its  surface,  but  with  different  degrees  of 
angular  velocity  in  proportion  to  their  proximity ;  the  same  distance  of 
advance  A  B  subtending  a  greater  angle,  A  P  B  =  c  P  p,  at  the  near 
object  P  than  at  the  remote  one  Q. 

(69.)  A  consequence  of  the  familiar  appearance  we  have  adduced  in 
illustration  of  these  principles  is  worth  noticing,  as  we  shall  have  occa- 
sion to  refer  to  it  hereafter.  We  observe  that  every  object  nearer  to  us 
than  that  ou  which  our  eye  is  fixed  appears  to  recede,  and  those  farther 
from  us  to  advance  in  relation  to  one  another.  If  then  we  did  not  know, 
or  could  not  judge  by  any  other  appearances,  which  of  two  objects  were 
nearer  to  us,  this  apparent  advance  or  recess  of  one  of  them,  when  the 
eye  is  kept  steadily  fixed  on  the  other,  would  furnish  a  criterion.     In  9 


PARALLACTIC   MOTION. 


Wy 


dark  night,  for  instance,  when  all  intermediate  objects  are  unseen,  the 
apparent  relative  movement  of  two  lights  which  we  are  assured  are  them-  . 
selves  fixed,  will  decide  as  to  their  relative  proximities.   That  which  seems 
to  advance  with  us  and  gain  upon  the  other,  or  leave  it  behind  it,  h  the 
farthest  from  us. 

(70.)  The  apparent  angular  motion  of  an  object,  arising  from  a  change 
of  our  point  of  view,  is  called  in  general  parallax,  and  it  is  always  ex- 
pressed by  the  angle  APB  tuhtended  at  the  object  P  (see  fig.  of  art.  68) 
by  a  line  joining  the  two  points  of  view  A  B  under  consideration.  For 
it  is  evident  that  the  difference  of  angular  position  of  P,  with  respect  to 
the  invariable  direction  ABD,  when  viewed  from  A  and  from  B,  is  the 
difii  r'<;nce  of  the  two  angles  DBP  and  DAP ;  now,  DBP  being  the  exte- 
rior angle  of  the  triangle  ABP,  is  equal  to  the  sum  of  the  interior  and 
opposite,  DBP  =  DAP  +  APB,  whence  DBP  —  DAP  =  APB. 

(71.)  It  follows  from  what  has  been  said  that  the  amount  of  parallactic 
motion  arising  from  any  given  change  of  our  point  of  view  is,  cceteria 
paribus,  less,  as  the  distance  of  an  object  viewed  is  greater;  and  when 
that  distance  is  extremely  great  in  comparison  with  the  change  in  our 
point  of  view,  the  parallax  becomes  insensible ;  or,  in  other  words,  objects 
do  not  appear  to  vary  in  situation  at  all.  It  is  on  this  principle,  that  in 
alpine  regions  visited  for  the  first  time  we  are  surprised  and  confounded 
at  the  little  progress  we  appear  to  make  by  a  considerable  change  of 
place.  An  hour's  walk,  for  instance,  produces  but  a  small  parallactic 
change  in  the  relative  situations  of  the  vast  and  distant  masses  which 
surround  us.  Whether  we  walk  round  a  circle  of  a  hundred  yards  in 
diameter,  or  merely  turn  ourselves  round  in  its  centre,  the  distant  pano- 
rama, presents  almost  exactly  the  same  aspect, — we  hardly  seem  to  have 
changed  our  point  of  view. 

(72.)  Whatever  notion,  in  other  respects,  we  may  form  of  the  stars,  it 
is  quite  clear  they  must  be  immensely  distant.  Were  it  not  so,  the  appa- 
rent angular  ini'^rval  between  any  two  of  them  seen  over-head  would  be 
much  greater  than  when  seen  near  the  horizon,  and  the  constellations, 
instead  of  preserving  the  same  appearances  and  dimensions  during  their 
wholw  diurnal  course,  would  appear  to  enlarge  as  they  rise  higher  in  the 
sky,  as  we  see  a  small  cloud  in  the  horizon  swell  into  a  grefit  over- 
shadowing canopy  when  drifted  by  the  wind  across  our  zenith,  or  as  may 
be  seen  in  the  annexed  figure,  where  ah,  AB,  ab,  are  three  different 
positions  of  the  same  stars,  as  they  would,  if  near  the  earth,  be  seen 
from  a  spectator  S,  under  the  visual  angles  aSb,  ASB.  No  such  change 
of  apparent  dimension,  however,  is  observed.  The  nicest  measurements 
of  the  apparent  angular  distance  of  any  two  «tor«  inter  se,  taken  in  any 


56 


OUTLINES   OF  ASTROKOKiT. 

« 

.•  ,,-,':,:'K>n  Fig.  8.     i'-iy^  .'":•'■ 


parts  of  their  diurnal  coarse,  (after  allowing  for  the  unequal  effects  of 
refhustion,  or  when  taken  at  such  times  that  this  cause  of  distortion  shall 
act  equally  on  hoth,)  manifest  not  the  slightest  perceptible  variation. 
Not  only  this,  but  at  whatever  point  of  the  earth's  surface  the  measure- 
ment is  performed,  the  results  are  absolutely  identical.  No  instruments 
ever  yet  invented  by  man  are  delicate  enough  to  indicate,  by  an  increase 
or  diminution  of  the  angle  subtended,  that  one  point  of  the  earth  is  nearer 
to  or  further  from  the  stars  than  another. 

(73.)  The  necessary  conclusion  from  this  is,  that  the  dimensions  of 
the  earth,  large  as  it  is,  are  comparatively  nothing,  absolutely  impercep- 
tible, when  compared  with  the  interval  which  separates  the  stars  from  the 
earth.  If  an  observer  walk  round  a  circle  not  more  than  a  few  yards  in 
diameter,  and  from  different  points  in  its  circumference  measure  with  a 
sextant  or  other  more  exact  iiastrument  adapted  for  the  purpose,  the 
angles  PAQ,  PBQ,  PCQ,  subtended  at  those  stations  by  two  well-defined 
points  in  his  visible  horizon,  PQ,  he  will  at  once  be  advertised,  by  the 
difference  of  the  results,  of  his  change  of  distance  from  them  arising  from 
his  change  of  place,  although  that  difference  may  be  so  small  as  to  pro- 
duce no  change  in  their  general  aspect  to  his  unassisted  sight.  This  is 
one  of  the  innumerable  instances  where  accurate  measurement  obtained 
by  instrumental  means  places  us  in  a  totally  different  situation  iu  respect 
to  matters  of  fact,  and  conclusions  thence  deducible,  from  what  we  should 
hold,  were  we  to  rely  in  ii'i  cases  on  the  mere  judgment  of  the  eye.  To 
so  great  a  nicety  have  such  observations  been  carried  by  the  aid  of  an 
instrument  called  a  theodolite,  tbat  a  circle  of  the  diameter  above  men- 
tioned may  thus  be  rendered  sensible,  may  thus  be  detected  to  have  a 
size,  and  an  ascertainable  place,  by  reference  to  objects  distant  by  fully 
100,000  times  its  own  dimensions.  Observations,  differing,  it  is  time, 
somewhat  in  method,  but  identical  in  principle,  and  executed  with  quite 
as  much  exactness,  have  been  applied  to  the  stars,  and  with  a  result  such 
as  has  been  already  stated.     Hence  it  follows^  incontrovertibly,  that  the 


THB  DISTANCE  OF  THE  STARS  IMMENSE. 


57 


,■«■•  /" .  •• 


distance  of  the  stars  from  the  earth  cannot  be  so  small  as  100,000  of  the 
earth's  diameters.  It  is,  indeed,  incomparably  greater ;  for  we  shall  here- 
after find  it  fully  demonstrated  that  the  distance  just  named,  immense  as 
it  may  appear,  is  yet  much  underrated. 

(74.)  From  such  a  distance,  to  a  spectator  with  our  faculties,  and 
furnished  with  our  instruments,  the  earth  would  be  imperceptible ;  and, 
reciprocally,  an  object  of  the  earth's  size,  placed  at  the  distance  of  the 
stars,  would  be  equally  undiscemible.  If,  therefore,  at  the  point  on  which 
a  spectator  stands,  we  draw  a  plane  touching  the  globe,  and  prolong  it  in 
imagination  till  it  attain  the  region  of  the  stars,  and  through  the  centre 
of  the  earth  conceive  another  plane  parallel  to  the  former,  and  co-e'xtensivc 
with  it,  to  pass;  these,  although  separated  throughout  their  whole  extent 
by  the  same  interval,  viz.,  a  semi-diameter  of  the  earth,  will  yet,  on  ac- 
count of  the  vast  distance  at  which  that  interval  is  seen,  be  confounded 
together,  and  undistinguishable  from  each  other  in  the  region  of  the  stars, 
when  viewed  by  a  spectator  on  the  earth.  The  zone  they  there  include 
will  be  of  evanescent  breadth  to  his  eye,  pnd  will  only  mark  out  a  great 
circle  in  the  heavens,  one  and  the  same  for  both  the  stations.  This  great 
circle,  when  spoken  of  as  a  circle  of  the  sphere,  is  called  the  celestial  hori- 
zon or  simply  the  horizon,  and  the  two  planes  just  described  are  also  spoken 
of  as  the  sensible  and  the  rational  horizon  of  the  observer's  station. 

(75.)  From  what  has  been  said  (art.  73)  of  the  distance  of  the  stars, 
it  follows,  that  if  wo  suppose  a  spectator  at  the  centre  of  the  earth  to  have 
his  view  bounded  by  the  rational  horizon,  in  exactly  the  same  manner  as 
that  of  a  corresponding  spectator  on  the  surface  is  by  his  sensible  horizon, 
the  two  observers  will  see  the  same  stars  in  the  same  relative  situations, 
each  beholding  that  entire  bemisphore  of  the  heavens  which  is  above  the 
celestial  horizon^  corresponding  to  their  common  zenith.     Now,  so  far  of 


'  re 


OUTLINES   OP  ASTRONOMT. 


appearances  go,  it  is  clearly  the  same  thing  whether  the  heavens,  that  is, 
all  space,  with  its  contents,  revolve  round  a  spectator  at  rest  in  the  earth's 
centre,  or  whether  that  spectator  simply  turn  round  in  the  opposite  direc- 
tion in  his  place,  and  view  them  in  succession.  The  aspect  of  the  heavens, 
at  every  instant,  as  referred  to  his  horizon  (which  must  be  supposed  to 
turn  with  him),  will  be  the  same  in  both  suppositions.  And  since,  as  has 
been  shown,  appearances  are  also,  so  far  as  the  stars  are  concerned,  the 
same  to  a  spectator  on  the  surface  as  to  one  at  the  centre,  it  follows  that, 
whether  we  suppose  the  heavens  to  revolve  without  the  earth,  or  the  earth 
within  the  heavens,  in  the  opposite  direction,  the  diurnal  phenomenon,  to 
all  its  inhabitants,  will  be  no  way  different. 

(76.)  The  Copernican  astronomy  adopts  the  latter  as  the  true  explana- 
tion of  these  phenomena,  avoiding  thereby  tLo  necessity  of  otherwise  re- 
sorting to  the  cumbrous  mechanism  of  a  solid  but  invisible  sphere,  to 
which  the  stars  must  ^le  supposed  attached,  in  order  that  they  may  be 
carried  round  the  earth  without  derangement  of  their  relative  situations 
inter  se.  Such  a  contrivance  would,  indeed,  suffice  to  explain  the  diurnal 
revolution  of  the  stars,  so  as  to  "save  appearances;"  but  the  movements 
of  the  sun  and  moon,  as  well  as  those  of  the  planets,  are  incompatible  with 
such  a  supposition,  as  will  appear  when  we  come  to  treat  of  these  bodies. 
On  the  other  hand,  that  a  spherical  mass  of  moderate  dimensions  (or, 
rather,  when  compared  with  the  surrounding  and  visible  universe,  of  eva- 
nescent magnitude),  held  by  no  tie,  and  free  to  move  and  to  revolve,  should 
do  so,  in  conformity  with  those  general  laws  which,  so  far  as  we  know, 
regulate  the  motions  of  all  material  bodies,  b  so  far  from  being  a  postulate 
difficult  to  be  conceded,  that  the  wonder  would  rather  be  should  the  fact 
prove  otherwise.  As  a  postulate,  therefore,  we  shall  henceforth  regard  it; 
and  as,  in  the  progress  of  our  work,  analogies  offer  themselves  in  its  sup- 
port from  what  we  observe  of  other  celestial  bodies,  we  shall  not  fail  to 
point  them  out  to  the  reader's  notice. 

(77.)  The  earth's  rotation  on  its  axis  so  admitted,  explaining,  as  it  evi- 
dently does,  the  apparent  motion  of  the  stars  in  a  completely  satisfactory 
manner,  prepares  us  for  the  further  admission  of  its  motion,  bodily,  in 
space,  should  such  a  motion  enable  us  to  explain,  in  a  manner  equally  so, 
the  apparently  complex  and  enigmatical  motions  of  the  sun,  moon,  and 
planets.  The  Copernican  astronomy  adopts  this  idea  in  its  full  extent, 
ascribing  to  the  earth,  in  addition  to  its  motion  of  rotation  about  an  axis, 
aliio  one  of  translation  or  transference  through  space,  in  such  a  course  or 
orbit  J  and  so  regulated  in  direction  and  celerity,  as,  taken  in  conjunction 
with  the  motions  of  the  other  bodies  of  the  universe,  shall  render  a  ration- 


KBLATIVE  MOTION. 


59 


al  account  of  the  appearances  they  suooesBively  present, —  tbat  is  to  nay, 
an  account  of  which  the  several  parts,  postulates,  propositions,  dcdaotiou<!, 
intelligibly  cohere,  without  contradicting  each  other  or  the  nature  of  things 
as  concluded  from  experience.  In  this  view  of  the  Copernican  doctrine 
it  is  rather  a  geometrical  conception  than  a  physical  theory,  inasmuch  as  it 
simply  assumes  the  requisite  motions,  without  attempting  to  explain  their 
mechanical  origin,  or  assign  them  any  dependence  on  physical  causes.  The 
Newtonian  theory  of  gravitation  supplies  this  deficiency,  and,  by  showing 
that  all  the  motions  required  by  the  Copernican  conception  rmist,  and  thai 
no  others  can,  result  from  a  single,  intelligible,  and  very  simple  dynamical 
law,  has  given  a  degree  of  certainty  to  this  conception,  as  a  matter  of  fact, 
which  attaches  to  no  other  creation  of  the  human  mind. 

(78.)  To  understand  this  conception  in  its  further  developments,  the 
reader  must  bear  steadily  in  mind  the  distinction  between  relative  and  ab- 
solute motion.  Nothing  is  easier  to  perceive  than  that,  if  a  spectator  at 
rest  view  a  certain  number  of  moving  objects,  they  will  group  and  arrange 
themselves  to  his  eye,  at  each  successive  moment,  in  a  very  difierent  way  from 
what  they  would  do  were  he  in  active  motion  among  them, —  if  he  formed 
one  of  them,  for  instance,  and  joined  in  their  dance.  This  is  evident  from 
what  has  been  said  before  of  parallactic  motion ;  but  it  will  be  asked.  How 
is  such  a  spectator  to  disentangle  from  each  other  the  two  parts  of  the 
apparent  motions  of  these  external  objects,  —  that  which  arises  from  the 
effect  of  his  own  change  of  place,  and  which  is  therefore  only  apparent 
(or,  as  a  German  metaphysician  would  say,  subjective  —  having  reference 
only  to  him  as  perceiving  it),  —  and  that  which  is  real  (or  objective  —  hav- 
ing a  positive  existence,  whether  perceived  by  him  or  not)  ?  By  what 
rule  is  he  to  ascertain,  from  the  appearances  presented  to  him  while  him- 
self in  motion,  what  would  be  the  appearances  were  he  at  rest  ?  It  by 
no  means  follows,  indeed,  that  he  would  even  then  at  once  obtain  a  clear  con- 
ception of  all  the  motions  of  all  the  objects.  The  appearances  so  presented 
to  him  would  have  still  something  subjective  about  them.  They  would  be 
still  appearances,  not  geometrical  realities.  They  would  still  have  a  refe- 
rence to  the  point  of  view,  which  might  be  very  unfavourably  situated 
(jis,  indeed,  is  the  case  in  our  system)  for  affording  a  clear  notion  of  the 
real  movement  of  each  object.  No  geometrical  figure,  or  curve,  is  seen 
by  the  eye  as  it  is  conceived  by  the  mind  to  exist  in  reality.  The  laws 
of  perspective  interfere  and  alter  the  apparent  directions  and  foreshorten 
the  dimensions  of  its  several  parts.  If  the  spectator  be  unfavourably 
situated,  as,  for  instance,  nearly  in  the  plane  of  the  figure  (which  is  tho 
case  we  have  to  deal  with),  they  may  do  so  to  such  an  extent,  as  to  make 


M 


60 


OUTLINES  OP  ASTRONOMY. 


a  considerable  effort  of  imagination  necessary  to  pass  from  the  sensible  to 
the  real  form. 

(79.)  Still,  preparatory  to  this  ultimate  step,  it  is  first  necessary  that 
the  spectator  should  free  or  clear  the  appearances  from  the  disturbing 
influence  of  his  own  change  of  place.  And  this  he  can  always  do  by  the 
following  general  rule  or  proposition :  — 

The  relative  motion  of  two  bodies  is  the  same  as  if  either  of  them 
were  at  rest,  and  all  its  motion  communicated  to  the  other  in  an  opposite 
direction} 

Hence,  if  two  bodies  move  alike,  they  will,  when  seen  from  each  other 
(without  reference  to  other  near  bodies,  but  only  to  the  starry  sphere), 
appear  at  rest.  Hence,  also,  if  the  absolute  motions  of  two  bodies  be 
uniform  and  rectilinear,  their  relative  motion  is  so  also. 

(80.)  The  stars  are  so  distant,  that  as  we  have  seen  it  is  absolutely 
indifferent  from  what  point  of  the  earth's  surface  we  view  them.  Their 
configurations  inter  se  are  identically  the  same.  It  is  otherwise  with  the 
sun,  moon,  and  planets,  which  are  near  enough  (especially  the  moon)  to 
be  parallactically  displaced  by  change  of  station  from  place  to  place  on 
one  globe.  In  order  that  astronomers  residing  on  different  points  of  the 
earth's  surface  should  be  able  to  compare  their  observations  with  effect,  it 
is  necessary  that  they  should  clearly  understand  and  take  account  of  this 
etiect  of  the  difference  of  their  stations  on  the  appearance  of  the  outward 
universe  as  seen  from  each.  As  an  exterior  object  seen  from  one  would 
appear  to  have  shifted  its  place  were  the  spectator  suddenly  transported  to 
the  other,  so  two  spectators,  viewing  it  from  the  two  stations  at  the  same 
instant,  do  not  see  it  in  the  same  direction.  Hence  arises  a  necessity  for 
the  adoption  of  a  conventional  centre  of  reference,  or  imaginary  station 
of  observation  common  to  all  the  world,  to  which  each  observer,  wherever 
situated,  may  refer  (or,  as  it  is  called,  reduce)  his  observations,  by  calcu- 
lating and  allowing  for  the  effect  of  his  local  position  with  respect  to  that 
common  centre  (supposing  him  to  possess  the  necessary  data).  If  there 
were  only  two  observers,  in  fixed  stations,  one  might  agree  to  refer  his 
observations  to  the  other  station ;  but,  as  every  locality  on  the  globe  may 
be  a  station  of  observation,  it  is  far  more  convenient  and  natural  to  fix 

*  This  proposition  is  equivalent  to  the  following,  which  precisely  meets  the  case  pro- 
posed, but  requires  somewhat  more  thought  for  its  clear  apprehension  than  can  perhaps 
be  expected  from  a  beginner :  — 

Prof. — If  two  bodies,  A  and  B,  be  in  motion  independently  of  each  other,  the  motion 
which  B  teen  from  A  would  appear  to  have  if  A  were  at  rett  it  the  tame  with  that  which 
It  iDould  appear  to  have,  A  being  in  motion,  if,  in  addition  to  itt  oum  motion,  a  motion 
equal  to  A' t  and  in  the  tame  direction  were  communicated  to  it.       r^  t 

'^fi  /  /  ' '  .  ■        ■ 


BBLATIVB  MOTION. 


ei 


8  the  case  pro- 
an  can  perhaps 


upon  a  point  eqoally  related  to  all,  as  the  common  point  of  reference ', 
and  this  can  be  no  other  than  the  centre  of  the  globe  itself.  The  paral- 
lactic change  of  apparent  place  which  would  arise  in  an  object,  could  any 
observer  suddenly  transport  himself  to  the  centre  of  the  earth,  is  evidently 
the  angle  0  S  P,  subtend.  I  on  the  object  S  by  that  radius  0  P  of  the 
earth  which  joins  the  centre  and  the  place  P  of  observation. 


:.A    ?'-\"- 


Fig.  10. 


M 


n  each  other 
irry  sphere), 
0  bodies  be 


'* 


•)     •;»• 


her,  the  motion 
with  that  which 
otion,  a  motion 


'  .".vU 


63 


OUTLINES  OF  ASTKOMOMT. 


CHAPTER  II. 

TEnMINOLOOY  AND  ELSMENTART  OEOMETRICAL  CONCEPTIONS  AND 
RELATIONS. — TERMINOLOQY  RELATING  TO  TUE  GLOBE  OF  THE 
EARTH  —  TO  THE  CELESTIAL  SPHERE.  —  CELESTIAL  PERSPECTIVE. 


(81.)  Several  of  the  terms  in  use  among  astronomers  have  been  ex- 
plained in  the  preceding  chapter,  and  others  used  anticipatively.  But  the 
technical  language  ol'  every  subject  requires  to  be  formally  («tated,  both 
for  consistency  of  usage  and  definiteness  of  conception.  We  aball  there- 
fore proceed,  in  the  first  place,  to  define  a  number  of  terms  in  perpetual 
use,  having  relation  to  the  globe  of  the  earth  and  the  celestial  sphere. 

Definition  1.  The  axis  of  the  earth  is  that  diameter  about  which 
it  revolves,  with  a  uniform  motion,  from  west  to  east;  performing  one 
revolution  in  the  interval  which  elapses  between  any  star  leaving  a  cer- 
tain point  in  the  heavens,  and  returning  to  the  same  point  again. 

(83.)  Def.  2.  The  poles  of  the  earth  are  the  points  where  its  axis 
meets  its  surface.  The  North  Pole  b  that  nearest  to  Europe;  the  South 
I'ole  that  most  remote  from  it. 

(84.)  Def.  S.  The  earth's  equator  is  a  great  circle  on  its  surface, 
equidistant  from  its  poles,  dividing  it  into  two  hemispheres  —  a  northern 
and  a  southern;  in  the  midst  of  which  are  situated  the  respective  poles 
of  the  earth  of  those  names.  The  plajie  of  the  equator  is,  therefore,  a 
plane  perpendicular  to  the  earth's  axis,  and  passing  through  its  centre. 

(85.)  Def.  4.  The  terrestrial  meridian  of  a  station  on  the  earth's 
sunace,  is  a  great  circle  of  the  globe  passing  through  both  poles  and 
th  -ough  the  plane.  The  plane  of  the  meridian  is  the  plane  in  which 
that  circle  lies. 

(86.)  Def.  5.  The  sensible  and  the  rational  horizon  of  any  station 
have  been  already  defined  in  art.  74. 

(87.)  Def.  6.  A  meridian  line  is  the  line  of  intersection  of  the 
plane  of  the  meridian  of  any  station  with  the  plane  of  the  sensible 
horizon,  and  therefore  marks  the  north  and  south  points  of  the  horizon, 
or  the  directions  in  which  a  spectator  must  set  out  if  he  would  travel 
directly  towards  the  north  or  south  pole. 


TBRMINOLOOT. 


68 


(88.)  DBF.  7.  Thfc  latitude  of  a  place  on  the  earth '«  surface  is  its 
angular  distance  from  the  equator,  measured  on  its  own  terrestrial  meri- 
dian :  it  is  reckoned  in  degrees,  minutes,  and  seconds,  from  0  up  to  00", 
and  northwards  or  southwards  according  to  the  hemisphere  the  place  lies 
in.  Thus,  the  observatory  at  Orecnwich  is  situated  in  51"  28'  40"  north 
latitude.  This  definition  of  latitude,  it  will  be  observed,  is  to  be  con- 
sidered as  only  temporary.  A  more  exact  knowledge  of  the  physical 
structure  and  figure  of  the  earth,  aud  a  better  acquaintance  with  the 
niceties  of  astronomy,  will  render  some  modification  of  its  terms,  or  a 
different  manner  of  considering  it,  necessary. 

(80.)  Dkp.  8.  Paralleh  of  latitude  are  small  circles  on  the  earth's 
surface  parallel  to  the  equator.  Every  point  in  such  a  circle  has  thcT 
same  latitude.  Thus,  Greenwich  is  said  to  be  situated  in  the  parallel  of 
6P28'40". 

(90.)  Def.  9.  The  longitude  of  a  place  on  the  earth's  surface  is  the 
inclination  of  its  meridian  to  that  of  some  fixed  station  referred  to  as  a 
point  to  reckon  from.  English  astronomers  and  geographers  use  the  ob- 
servatory at  Greenwich  for  this  station ;  foreigners,  the  principal  observa- 
tories of  their  respective  nations.  Some  geographers  have  adopted  the 
island  of  Ferro.  Hereafter,  when  we  speak  of  longitude,  we  reckon  froa 
Greenwich.  The  longitude  of  a  place  is,  therefore,  measured  by  the  arc 
of  the  equator  intercepted  between  the  meridian  of  the  place  and  that  of 
Greenwich;  or,  which  is  the  same  thing,  by  the  spherical  angle  at  the 
pole  included  between  these  meridians. 

(91.)  As  latitude  is  reckoned  north  or  south,  so  longitude  is  usually 
said  to  be  reckoned  west  or  east.  It  would  add  greatly,  however,  to  sys- 
tematic regularity,  and  tend  much  to  avoid  confusion  and  ambiguity  in 
computations,  were  this  mode  of  expression  abandoned,  and  longitudes 
reckoned  invariably  westward  from  their  origin  round  the  whole  circle 
from  0  to  360°.  Thus,  the  longitude  of  Paris  is,  in  coiP"  u>"  parlance, 
either  2°  20'  22"  east,  or  357°  39'  38"  west  of  Greenwich,  iiut,  in  the 
sense  in  which  we  shall  henceforth  use  and  recommend  others  to  use  the 
term,  the  latter  is  its  proper  designation.  Longitude  h  also  reckoned  in 
time  at  the  rate  of  24  h.  for  360°,  or  15°  per  ho'u*  En  this  system  the 
longitude  of  Paris  is  23  h.  50m.  38^8.' 

(92.)  Knowing  the  longitude  and  latitude  of  a  place,  it  may  be  laid 
down  on  an  artificial  globe;  and  thus  a  map  of  the  earth  may  be  con- 


'  To  distinguish  minutes  and  seconds  of  time  from  those  of  angular  measure  we 
■hall  invariably  adhere  to  the  distinct  system  of  notation  here  adopted  (°  ' ",  and  h.  m. 
8.)  Great  confusion  sometimes  arises  from  the  practice  of  using  the  same  marks 
for  both. 


^ 


OUTLINES  OF  ASTRONOMY. 


structcd.  Maps  of  particular  countries  are  detached  portions  of  this 
general  map,  extended  into  planes;  or  rather,  they  are  representations 
on  planes  of  such  portions,  executed  according  to  certain  conventic^nal 
systems  of  rules,  called  projections,  the  object  of  which  is  either  to 
distort  as  little  as  possible  the  outlines  of  countries  from  what  they  are 
on  the  globe — or  to  establish  easy  means  of  ascertaining,  by  inspection  or 
graphical  measurement,  the  latitudes  and  longitudes  of  places  which 
occur  in  them,  without  referring  to  the  globe  or  to  books  —  or  for  other 
peculiar  uses.     See  Chap.  IV. 

(93.)  Def.  10.  The  Tropics  are  two  parallels  of  latitude,  one  on  the 
north  and  the  other  on  the  south  side  of  the  equator,  over  every  point 
of  which  respectively,  the  sun  in  its  diurnal  course  passes  vertically  on 
the  21st  of  March  and  the  21st  of  September  in  every  year.  Their 
latitudes  are  about  23°  28'  respectively,  north  and  south. 

(94.)  Def.  11.  The  Arctic  and  Antarctic  circles  are  two  small  circles 
or  parallels  of  latitude  as  distant  from  the  north  and  south  poles  as  the 
tropics  are  from  the  equator,  that  is  to  say,  about  23°  28';  their  latitudes,' 
therefore,  are  about  66°  32'.  We  say  about,  for  the  places  of  these 
circles  and  of  the  tropics  are  continually  shifting  on  the  earth's  surface, 
though  with  extreme  slowness,  as  will  be  explained  in  its  proper  place. 

(95.)  Def.  12.  The  sphere  of  the  heavens  or  of  the  stars  is  au  ima- 
ginary spherical  surface  of  infinite  radius,  having  the  eye  of  any  specta- 
tor for  its  centre,  and  which  may  be  conceived  as  a  ground  on  which  the 
stars,  planets,  &c.,  the  visible  contents  of  the  universe,  are  seen  projected 
as  in  a  vast  picture.' 

(96.)  Def.  13.  The  ^ofes  of  the  celestial  sphere  are  the  points  of  that 
imaginary  sphere  towards  which  the  earth's  axis  is  directed. 

(97.)  Def.  14.  The  celestial  equator,  or,  as  it  is  often  called  by  as- 

'  The  ideal  sphere  without  us,  to  which  we  refek  the  places  of  objects,  and  which 
we  carry  along  with  us  wherever  we  go,  is  no  doubt  intimately  connected  by  associa- 
ti'^n,  if  not  entirely  dependent  on  that  obscure  perception  of  sensation  in  the  retineB  of 
our  eyes,  of  which,  even  when  closed  and  unexcited,  we  cannot  entirely  divest  them. 
We  have  a  real  spherical  surface  within  our  eyes,  the  seat  of  sensation  and  vision, 
corresponding,  point  for  point,  to  the  external  sphere.  On  this  the  stars,  &.c.  are  really 
mapped  down,  as  we  have  supposed  them  in  the  text  to  be,  on  the  imaginary  concave 
of  the  heavens.  When  the  whole  surface  of  the  retina  is  excited  by  light,  habit  leads 
us  to  associate  it  with  the  idea  of  a  real  surface  existing  without  us.  Thus  we  become 
impressed  with  the  notion  of  a  $ky  and  a  heaven,  but  the  concave  surface  of  the  retina 
itself  is  the  true  seat  of  all  visible  angular  dimension  and  angular  motion.  The  sub- 
stitution of  the  retina  for  the  heavens  would  be  awkward  and  inconvenient  in  language, 
but  it  may  always  be  mentally  made.  (See  Schiller's  pretty  enigma  on  the  eye,  in.  'ail 
Turandot.)  >  h 


I 


TERMINOLOGY. 


65 


tronomers,  the  eqvinoctial,  is  a  great  circle  of  the  celestial  sphere,  marked 
out  by  the  indefinite  extension  of  the  plane  of  the  terrestrial  equator. 

(98.)  Dep.  15.  The  celestial  horizon  of  any  place  is  a  great  circle  of 
the  sphere  marked  out  by  the  indefinite  extension  of  the  plane  of  any 
spectator's  sensible  or  (which  comes  to  the  same  thing  as  will  presently 
be  shown,)  his  rational  horizon,  as  in  the  ease  of  the  equator. 

(99.)  Def.  16.  The  zenith  and  nadir^  of  a  spectator  are  the  two 
points  of  the  sphere  of  the  heavens,  vertically  over  his  head,  and  verti- 
cally under  his  feet,  or  the  poles  of  the  celestial  horizon ;  that  is  to  say, 
points  90°  distant  from  every  point  in  it. 

(100.)  Dep.  17.  Vertical  circles  of  the  sphere  are  great  circles  passing 
through  the  zenith  and  nadir,  or  great  circles  perpendicular  to  the  horizon. 
On  these  are  measured  the  altitudes  of  objects  above  the  hori/on  —  the 
complements  to  which  are  their  zenith  distances. 

(101.)  Dep,  18.  The  celestial  meridian  of  a  spectator  is  the  great  circle 
marked  out  on  the  sphere  by  the  prolongation  of  the  plane  of  his  terres- 
trial meridian.  If  the  earth  be  supposed  at  rest,  this  is  a  fixed  circle,  and 
all  the  stai-s  are  carried  across  it  in  their  diurnal  courses  from  east  to  west. 
If  the  stars  rest  and  the  earth  rotate,  the  spectator's  meridian,  like  his 
horizon  (art.  52),  sweeps  daily  across  the  stars  from  west  to  east.  When- 
ever in  future  we  speak  of  the  meridian  of  a  spectator  or  observer,  we 
intend  the  celestial  meridian,  which  being  a  circle  passing  through  the 
poles  of  the  heavens  and  the  zenith  of  the  observer,  is  necessarily  a  verti- 
cal circle,  and  passes  through  the  north  and  south  points  of  the  horizon. 

(102.)  Dep.  19.  The^r«'?ne  vertical  is  a  vertical  circle  perpendicular  to 
the  meridian,  and  which  therefore  passes  through  the  east  and  west  points 
of  the  horizon. 

(103.)  Def.  20.  Azimuth  is  the  angular  distance  of  a  celestial  object 
from  the  north  or  south  point  of  the  horizon  (according  as  it  is  the  north 
or  south  pole  which  is  elevated),  when  the  object  is  referred  to  the  horizon 
by  a  vertical  circle ;  or  it  is  the  angle  comprised  between  two  vertical 
planes  —  one  passing  through  the  elevated  pole,  the  other  through  the 
object.  Azimuth  may  be  reckoned  eastwards  or  westwards,  from  the 
north  or  south  point,  and  is  usually  so  reckoned  only  to  180°  either  way. 
But  to  avoid  confusion,  and  to  preserve  continuity  of  interpretation  when 
algebraic  symbols  are  used  (a  point  of  essential  importance,  hitherto  too 
little  insisted  on),  we  shall  always  reckon  azimuth  from  the  points  of  the 
horizon  most  remote  from  the  elevated  pole,  westward  (so  as  to  agree  in 
general  directions  with  the  apparent  diurnal  motion  of  the  stars),  and 


'  From  Arabic  words, 
whence  our  nether. 

5 


Nadir  corresponds  evidently  to  the  German  nieder,  (down 


66 


OUTLINES   OF  ASTRONOMY. 


'  / 


carry  its  reckoning  from  0°  to  360"  if  always  reckoned  positive,  consider- 
ing the  eastward  reckoning  as  negative. 

(104.)  Def.  21.  The  altitude  of  a  heavenly  body  is  its  apparent  angular 
elevation  above  the  horizon.  It  is  the  complement  to  90°,  therefore,  of 
its  zenith  distance.  The  altitude  and  azimuth  of  an  object  being  known, 
its  place  in  the  visible  heavens  is  determined. 

(105.)  Def.  22.  The  declination  of  a  heavenly  body  is  its  angular 
distance  from  the  equinoctial  or  celestial  equator,  or  the  complement  to 
90°  of  its  angular  distance  from  the  nearest  pole,  which  latter  distance  is 
called  its  Polar  distance.  Declinations  are  reckoned  plus  or  minus, 
according  as  the  object  is  situated  in  the  northern  or  southern  celestial 
hemisphere.  Polar  distances  are  always  reckoned  from  the  North  Pole, 
from  0°  up  to  180",  by  which  all  doubt  or  ambiguity  of  expression  with 
respect  to  sign  is  avoided. 

(106.)  Def.  23.  Hour  circles  of  the  sphere,  or  circles  of  declination, 
are  great  circles  passing  through  the  poles,  and  of  course  perpendicular  to 
the  equinoctial.  The  hour  circle,  passing  through  any  particular  heavenly 
body,  serves  to  refer  it  to  a  point  in  the  equinoctial,  as  a  vertical  circle 
does  to  a  point  in  the  horizon. 

(107.)  Def.  24.  The  hour  angle  of  a  heavenly  body  is  the  angle  at 
the  pole  included  between  the  hour  circle  passing  through  the  body,  and 
the  celestial  meridian  of  the  place  of  observation.  We  shall  always 
reckon  it  positively  from  the  upper  culmination  (art.  125)  westwards,  or 
in  conformity  with  the  apparent  diurnal  motion,  completely  round  the 
circle  from  0"  to  360°.  ITour  angles,  generally,  are  angles  included  at 
the  pole  between  different  hour  circles. 

(108.)  Def.  25.  The  rifflu  ascension  of  a  heavenly  body  is  the  arc  of 
the  equinoctial  included  between  a  certain  point  in  that  circle  called  the 
Vernal  Equinox^  and  the  point  in  the  same  circle  to  which  it  is  referred 
by  the  circle  of  declination  passing  through  it.  Or  it  is  the  angle  included 
between  two  hour  circles,  one  of  which  passes  through  the  vernal  equinox 
(and  is  called  the  equin  tial  colure),  the  other  through  the  body.  How 
the  place  of  this  initial  point  on  the  equinoctial  is  determined,  will  be 
explained  further  on. 

(109.)  The  right  ascensions  of  celestial  objects  are  always  reckoned 
■  eastwards  from  the  equinox,  and  are  estimated  either  in  degrees,  minutes, 
and  seconds,  as  in  the  case  of  terrestrial  longitudes,  from  0°  to  360°, 
which  completes  the  circle;  or,  in  time,  in  hours,  minutes,  and  seconds, 
from  Oh.  to  24h.  The  apparent  diurnal  motion  of  the  heavens  being 
contrary  to  the  real  motion  of  the  earth,  this  is  in  conformity  with  the 
westward  reckoning  of  longitudes.  (Art.  91.)  ' ' 


\  i 


■1 1 


TERMINOLOGY. 


67 


lis  or  minus. 


(110.)  Sidereal  time  is  reckoned  by  the  diurnal  motion  of  the  stars, 
or  rather  of  that  point  in  the  equinoctial  from  which  right  ascensions  are 
reckoned.  This  point  may  be  considered  as  a  star,  though  no  star  is,  in 
fact,  there;  and,  moreover,  the  point  itself  is  liable  to  a  certain  slow 
variation,  —  so  slow,  however,  as  not  to  affect,  perceptibly,  the  interval, 
of  any  two  of  its  successive  returns  to  the  meridian.  This  interval  is 
called  a  sidereal  day,  and  is  divided  into  24  sidereal  hours,  and  these  again 
into  minutes  and  seconds.  A  clock  which  marks  sidereal  time,  i.  e.  which 
goes  at  such  a  rate  as  always  to  show  Oh.  Om.  Os.  when  the  equinox  comes  on 
the  meridian,  is  called  a  sidereal  clock,  and  is  an  indispensable  piece  of  furni- 
ture in  every  observatory.  Hence  the  hour  angle  of  an  object  reduced  to 
time  at  the  rate  of  15°  per  hour,  expresses  the  interval  of  sidereal  time 
by  which  (if  its  reckoning  be  positive)  it  has  past  the  meridian ;  or,  if 
negative,  the  time  it  wants  of  arriving  at  the  meridian  of  the  place  of 
observation.  So  also  the  right  ascension  of  an  object,  if  converted  into 
time  at  the  same  rate  (since  360°  being  described  uniformly  in  24  hours, 
15°  must  be  so  described  in  1  hour),  will  express  the  interval  of  sidereal 
time  which  elapses  from  the  passage  of  the  vernal  equinox  across  the 
meridian  to  that  of  the  object  next  subsequent. 

(111.)  As  a  globe  or  maps  may  be  made  of  the  whole  or  particular 
regions  of  the  surface  of  the  earth,  so  also  a  globe,  or  general  map  of  the 
heavens,  as  well  as  charts  of  particular  parts,  may  be  constructed,  and  the 
stars  laid  down  in  their  proper  situations  relative  to  each  other,  afid  to 
the  poles  of  the  heavens  and  the  celestial  equator.  Such  a  representa- 
tion, once  made,  will  exhibit  a  true  appearance  of  the  stars  as  they 
present  themselves  in  succession  to  every  spectator  on  the  surface,  or  as 
they  may  be  conceived  to  be  seen  at  once  by  one  at  the  centre  of  the 
globe.  It  is,  therefore,  independent  of  all  geographical  localities.  There 
will  occur  in  such  a  representation  neither  zenith,  nadir,  nor  horizon  — 
neither  east  nor  west  points ;  and  although  great  circles  may  be  drawn  on 
it  from  pole  to  pole,  corresponding  to  terrestrial  meridians,  they  can  no 
longer,  in  this  point  of  view,  be  regarded  as  the  celestial  meridians  of 
fixed  points  on  the  earth's  surface,  since,  in  the  course  of  one  diurnal 
revolution,  every  point'  in  it  passes  beneath  each  of  them.  It  is  on 
account  of  this  change  of  conception,  and  with  a  view  to  establish  a  com- 
plete distinctic  1  between  the  two  branches  of  Geography  and  Uranogra- 
pliy^  that  astronomers  have  adopted  different  terms,  (viz.  declination  and 
right  ascension)  to  represent  those  arcs  in  the  heavens  which  correspond 
to  latitudes  and  longitudes  on  the  earth.     It  is  for  this  reason  that  thej 

-     •■     '  >    .-f     > - 

*  Ti?,  the  earth ;  ypa<puv,  to  describe  or  represent ;  ovpavos,  the  heavisn.    .■ 


i 


68 


\  ( 


OUTLINES   OF  ASTRONOMY. 


term  the  equator  of  the  heavens  the  equinoctial;  that  what  are  meridians 
on  the  earth  are  called  hour  circhs  in  the  heavens,  and  the  angles  they 
include  between  them  at  the  poles  are  called  hour  angles.  All  this  is 
convenient  and  intelligible ;  and  had  they  been  content  with  this  nomen- 
clature, no  confusion  could  ever  have  arisen.  Unluckily,  the  early 
astronomers  have  employed  ahc  ;ie  words  latitude  and  longitude  in  their 
uranography,  in  speaking  of  pvcs  of  circles  not  corresponding  to  those 
meant  by  the  same  words  on  the  ea  .,b,  but  having  reference  to  the  motion 
of  the  sun  and  planets  among  the  stars.  It  is  now  too  late  to  remedy 
this  confusion,  which  is  ingrafted  into  every  existing  work  on  astronomy : 
we  can  only  regret,  and  warn  the  reader  of  it,  that  he  may  be  on  his 
guard  when,  at  a  more  advanced  period  of  our  work,  we  shall  have  occa- 
sion to  define  and  use  the  terms  in  their  celestial  sense,  at  the  same  time 
urgently  recommending  to  future  writers  the  adoption  of  others  in  their 
places. 

(112.)  It  remains  to  illustrate  these  descriptions  by  reference  to  a 
figure.     Let  C  be  the  centre  of  the  earth,  N  C  S  its  axis;  then  are  N' 


iU>./\ 


Fig.  11. 


/ 


and  S  its  poles;  E  Q  its  equator;  A  B  the  parallel  of  latitude  of  th", 
station  A  on  its  surface ;  A  P  parallel  to  S  C  N,  the  direction  in  which 
an  observer  at  A  will  see  the  elevated  pole  of  the  heavens ;  and  A  Z,  the 
prolongation  of  the  terrestrial  radius  C  A,  that  of  his  zenith.  N  A  E  S 
will  be  his  meridian ;  N  G  S  that  of  some  fixed  station,  as  Greenwich ; 
and  G  E,  or  the  spherical  angle  G  N  E,  his  longitude,  and  E  A  his  lati- 
tude.    Moreover,  if  n  s  be  a  plane  touching  the  surface  in  A,  this  will 


f*lii- 


TERMINOLOGY. 


69 


be  his  sensible  horizon :  n  A  «  marked  on  that  plane  by  its  intersection 
with  his  meridian  will  be  his  meridian  line,  and  n  and  s  the  north  and 
south  points  of  his  horizon. 

(113.)  Again,  neglecting  the  size  of  the  earth,  or  conceiving  him 
stationed  at  its  centre,  and  referring  every  thing  to  his  rational  horizon  j 
let  the  annexed  figure  represent  the  sphere  of  the  heavens;  C  the  specta- 
tor; Z  his  zenith;  and  N  his  nadir:  then  will  H  A  0,  a  great  circle  of 
the  sphere,  whose  poles  are  ^  N,  be  his  celestial  horizon;  P  p  the 
elevated  and  depressed  POLES  of  the  heavens ;  H  P  the  altitude  of  the 
pole,  and  H  P  Z  E  0  his  meridian;  E  T  Q,  a  great  circle  perpendicular 
to  Pj  will  be  the  equinoctial;  and  if  T  represent  the  equinox,  r  T  will 
be  the  riglit  ascension,  T  S  the  declination,  and  P  S  the  polar  distance 
of  any  star  or  object  S,  referred  to  the  equinoctial  by  the  h^ur  circle  P 
S  Tp;  and  B  S  D  will  be  the  diurnal  circle  it  will  appear  to  describe 

i:,-   ,.,'      ■,:':  .r.    Fig.  12.    ,.,,      ■  ;,..V      ■>,;,/ 

..it  "' 


.     -  *      'i ."  f      "■'-J  ' 


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11  V--'' 

ly 

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k1 

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V^ 

— "  "^^  / 

/^  7 

v 

r 

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<r^ 

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JP 

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O 


'■*?. 


N 


about  the  pole.  Again,  if  we  refer  it  to  the  horizon  by  the  vertical  drrU 
Z  S  M,  0  M  will  be  its  azimuth,  M  S  its  altitude,  and  Z  !S  its  zenith 
distance.  H  and  0  are  the  north  and  south,  e  w  the  easf,  and  west  points 
of  his  horizon,  or  of  the  heavens.  Moreover,  if  H  /,  0  o,  be  small 
circles,  or  parallels  of  declination,  touching  the  horizon  in  its  north  and 
Bouth  points,  H  h  will  be  the  circle  cf  perpetual  apparition,  between 
which  and  the  elevated  pole  the  stars  never  set ;  0  o  that  of  perpetual 
ocadtation,  between  which  and  the  depressed  pole  they  never  rise.  In 
all  the  zone  of  the  heavens  between  H  It,  and  0  c,  they  rise  and  set;  any 
one  of  them,  as  S,  remaining  above  the  horizon  in  that  part  of  its  diurnal 
circle  represented  by  a  B  A,  and  below  it  throuj];hout  all  Ihe  part  repre- 
sented by  A  D  a.     It  will  exercise  the  reader  to  construct  this  figure  for 


:.^*i|y: 


'    y: 


10 


OUTLINES   OF  ASTRONOMY. 


I 


1 1 


several  different  elevations  of  the  jjole,  and  for  a  variety  of  positions  of 
the  star  S  in  each. 

(114.)  Celestial  pei^^pective  is  that  branch  of  the  general  scio'ioo  of 
perspective  which  teaches  us  to  conclude,  from  a  knowledge  of  the  n  nl 
situation  and  forma  of  objects,  lines,  angles,  motions,  &o.  with  n  :)pt:C;;  v) 
the  spectator,  their  apparent  aspects,  as  seen  by  him  p^-ojected  oc  t)ii? 
imaginary  concave  of  tho  heavens;  Rud,  vice  vemd,  frora  the  aj^parent 
configurations  and  movements  of  objects  •so  scon  projecteJ,  to  coi  .Jude, 
so  fir  as  they  can  be  thence  concluded,  their  real  geometrical  relations  to 
each  other  and  to  the  spectator.   It  agrees  with  ordinary  perspective  w^en 
only  a  small  visual  area  is  contemplated,  because  tho  concave  giound  of 
the  celtAjtiul  sphere,  for  a  sniall  extent,  may  be  regarded  as  t  plane  sur- 
face', <  u  which  objects  are  seen  projected  or  depicted  as  in  coinmou  j'er- 
spcctive.     But  wh  i;  large  amplitudes  of  the  visual  area  are  considered, 
or  when  the  whole  con  tern'  of  amct:  are  regarded  as  projected  on  the 
whole  interior  surface  of  f '"  .sphere,  it  becomes  necessary  to  use  a  different  > 
phraseology,  and  to  rcvM-t  to  a  different  form  of  conception.     In  common 
persptctive  there  is  a  single  "point  of  sight,"  or  "centre  of  ilw  picture," 
the  visual  line  from  the  eye  to  which  is  perpendicular  to  the  ''  plane  of 
the  picture,"  and  all  straight  lines  are  represented  by  straight  lines.     In 
celestial  perspective,  every  point  to  which  the  view  is  for  the  moment 
directed,  is  equally  entitled  to  be  considered  as  the  "centre  of  the  pic- 
ture," every  portion  of  the  surface  of  the  sphere  being  similarly  related 
to  the  eye.     Moreover,  every  straight  line  (supposed  to  be  indefinitely 
prol  (vjged)  is  projected  into  a  semicircle  of  the  sphere,  that,  namely,  in 
which  ^  plane  passing  through  the  line  and  the  eye  cuts  its  surface.    And 
every  system  of  parallel  straight  lines,  in  whatever  direction,  is  projected 
into  a  system  of  semicircles  of  the  sphere,  meeting  in  two  common  apexes, 
or  vanishing  points,  diametrically  opposite  to  each  other,  one  of  which 
corresponds  to  the  vanishing  point  of  parallels  in  ordinary  perspective ; 
the  other,  in  such  perspective  has  no  existence.     In  other  words,  every 
point  in  the  sphere  to  which  the  eye  is  directed  may  be  regarded  as  one 
of  the  vanishing  points,  or  one  apex  of  a  system  of  straight  lines,  parallel 
to  that  radius  of  the  sphere  which  passes  through  it,  or  to  the  direction 
of  the  line  of  sight,  seen  in  perspective  from  the  earth,  and  the  points 
diametrically  opposite,  or  that  from  which  he  is  looking,  as  the  other. 
And  any  great  circle  of  the  sphere  may  similarly  be  regarded  as  the 
vanishing  circle  of  a  system  of  planes,  parallel  to  its  own. 

(115.)  A  familiar  illustration  of  this  is  often  to  be  had  by  attending  to 
the  lines  of  light  seen  in  the  air,  when  the  sun's  rays  are  darted  through 
apertures  in  clouds,  the  sun  itself  being  at  the  time  obscured  behind  them. 


CELESTIAL  PERSPECTIVE. 


71 


These  lines  which,  marking  the  course  of  rays  emanating  from  a  point 
almost  infinitely  distant,  are  to  be  considered  as  parallel  straight  lines,  are 
thrown  into  great  circles  of  the  sphere,  having  two  apexes  or  points  of 
common  intersection  —  one  in  the  place  where  the  sun  itself  (if  not 
obscured)  would  be  seen.     The  other  diametrically  opposite.     The  first 

I,  ly  is  most  commonly  suggested  when  the  spectator's  view  is  towards  the 
t^un.  But  in  mountainous  countries^  the  phenomenon  of  sunbeams  con- 
verging towards  a  point  diametrically  opposite  to  the  sun,  and  as  much 
depressed  below  the  horizon  as  the  suu  is  elevated  above  it,  is  not  unfre- 

Miently  noticed,  the  back  of  the  spectator  being  turned  to  the  sun's  place. 
Occasionally,  but  much  more  rarely,  the  whole  course  of  such  a  system 
of  sunbeams,  stretching  in  semicircles  across  the  hemisphere  from  horizon 
to  horizon  (the  sun  being  near  setting),  may  be  seen.'  Thus  again,  the 
streamers  of  the  Aurora  Borealis,  which  are  doubtless  electrical  rays, 
parallel,  or  nearly  parallel  to  each  other,  and  to  the  dipping  needle,  usually 
appear  to  diverge  from  the  point  towards  which  the  needle,  freely  sus- 
pended, would  dip  northwards  {i.  e.  about  70"  below  the  horizon  and  23° 
west  of  north  from  London),  and  in  their  upward  progress  pursue  the 
course  of  great  circles  till  they  again  converge  (in  appearance)  towards 
the  point  diametrically  opposite  (i.  e.  70**  above  the  horizon,  and  23°  to  the 
eastward  of  south),  forming  a  sort  of  canopy  over-head,  having  that  point 
for  its  centre.  So  also  in  the  phenomenon  of  shooting  stars,  the  lines  of 
direction  which  they  appear  to  take  on  certain  remarkable  occasidns  of 
periodical  recurrence,  are  observed,  if  prolonged  backwards,  apparently  to 
meet  nearly  in  one  point  of  the  sphere ;  a  certain  indication  of  a  general 
near  approach  to  parallelism  in  the  real  directions  of  their  motions  on 
those  occasions.     On  which  subject  more  hereafter. 

(116.)  In  relation  to  this  idea  of  celestial  perspective,  we  may  conceive 
the  north  and  south  poles  of  the  sphere  as  the  two  vanishing  points  of  a 
system  of  lines  parallel  to  the  axis  of  the  earth ;  and  the  zenith  and  nadir 
of  those  of  a  system  of  perpendiculars  to  its  surface  at  the  place  of 
observation,  &c.  It  will  be  shown  that  the  direction  of  a  plumb-line,  at 
every  place  is  perpendicular  to  the  surface  of  still  water  at  that  place 

'  It  is  in  such  cases  only  that  we  conceive  them  as  circles,  the  ordinary  conventions 
of  plane  perspective  becoming  untenable.  The  author  had  the  good  fortune  to  witness 
on  one  occasion  the  phenomenon  described  in  the  text  under  circumstances  of  more 
than  usual  grandeur.  Approaching  Lyons  from  the  south  on  Sept.  30, 1826,  about  5^  h. 
p.  M.,  the  sun  was  seen  nearly  setting  behind  broken  masses  of  stormy  cloud,  from 
whose  apertures  streamed  forth  beams  of  rose-coloured  light,  traceable  all  across  the 
hemisphere  almost  to  their  opposite  point  of  convergence  behind  the  snowy  precipices 
of  Mont  Blanc,  conspicuously  visible  at  nearly  100  miles  to  the  eastward.  The  im- 
pression produced  was  that  of  another  but  feebler  sun  about  to  rise  from  behind  the 
mountain,  and  darting  forth  precursory  beams  to  meet  those  of  the  real  one  opposite. 


/ 


OUTLINES   OF  ASTRONOMY. 


which  is  the  true  horizon,  and  though  mathematically  speaking  no  two 
plumb-lines  are  exactly  parallel  (since  they  converge  to  the  earth's  centre), 
yet  over  very  small  tracts,  such  as  the  area  of  a  building  —  in  one  and 
the  same  town,  &o.,  the  difference  from  exact  parallelism  is  so  small  that 
it  may  be  practically  disregarded.'  To  a  spectator  looking  upwards  such 
a  system  of  plumb-lines  will  appear  to  converge  to  his  zenith ;  downwards, 
to  his  nadir.       i'   ■:'"■':::  .■■-/-  ,  ,■  ,         .    -..'-i^i^i^.-y-  '■^'^'-■..  ;r     ,- 

(117.)  So  also  the  celestial  equator,  or  the  equinoctial,  must  be  con- 
ceived as  the  vanishing  circle  of  a  system  of  planes  parallel  to  the  earth's 
o(juator,  or  perpendicular  to  its  axis.  The  celestial  horizon  of  any  spec- 
tator is  in  like  manner  the  vanishing  circle  of  all  planes  parallel  to  his 
true  horizon,  of  which  planes  his  rational  horizon  (passing  through  the 
earth's  centre)  is  one,  and  his  sensible  horizon  (the  tangent  plane  of  his 
station)  another.     . 

(118.)  Owing,  however,  to  the  absence  of  all  the  ordinary  indications 
of  distance  which  influence  our  judgment  in  respect  of  terrestrial  objects, 
owing  to  the  want  of  determinate  figure  and  magnitude  in  the  stars  and 
planets  as  commonly  seen  —  the  projection  of  the  celestial  bodies  on  the 
ground  of  the  heavenly  concave  is  not  usually  regarded  in  this  its  true 
light,  of  a  perspective  representation  or  picture,  and  it  even  requires  an 
effort  of  imagination  to  conceive  them  in  their  true  relations,  as  at  vastly 
different  distances,  one  behind  the  other,  and  forming  with  one  another 
lines  of  junction  violently  foreshortened,  and  including  angles  altogether 
differing  from  those  which  their  projected  representations  appear  to  make. 
To  do  so  at  all  with  effect  presupposes  a  knowledge  of  their  actual  situa- 
tions in  space,  which  it  is  the  business  of  astronomy  to  arrive  at  by  appro- 
priate considerations.  But  the  connections  which  subsist  among  the 
several  parts  0/  the  picture,  the  purely  geometrical  relations  among  the 
angles  and  sides  of  the  spherical  triangles  of  which  it  consists,^  constitute, 
under  the  name  of  Uranometry/^  a  preliminary  and  subordinate  branch  of 
the  general  science,  with  which  it  is  necessary  to  be  familiar  before  any 
further  progress  can  be  made.  Some  of  the  most  elementary  and  fre- 
quently occurring  of  these  relations  we  proceed  to  explaiii.  And  first,  as 
immediate  consequences  of  the  above  definitions,  the  following  propositions 
will  be  borne  in  mind. 

(119.)  The  altitude  of  the  elevated  pole  is  equal  to  the  latitude  of  the 
spectator's  geographical  station. 

For  it  appears,  see  fg.  art.  112,  that  the  angle  PAZ  between  the 

'  An  interval  of  a  mile  corresponds  to  a  convergence  of  plumb-lines  amounting  to 
comewhat  <es8  space  than  a  minute.  i 

3  Ovpavbf,  the  heavens ;  fitrfuv,  to  measure :  the  measurement  of  the  heavens. 


ELEMENTARY  EELATIONS. 


Ti 


73 


pole  and  the  zenith  is  equal  to  N  C  A,  and  the  angles  Z  A  n  and  N  C  E 
being  right  angles,  we  have  P  A  n=A  C  E.  Now  the  former  of  these 
is  the  elevation  of  the  pole  as  seen  from  E,  the  latter  is  the  angle  at  the 
earth's  centre  subtended  by  the  arc  E  A,  or  the  latitude  of  the  place. 

(120.)  Hence  to  a  spectator  at  the  north  pole  of  the  earth,  the  north 
pole  of  the  heavens  is  in  his  zenith.  As  he  travels  authward  it  becomes 
less  and  loss  elevated  till  he  reaches  the  equator,  when  both  poles  are  in 
his  horizon  —  south  of  the  equator  tii9  north  pole  becomes  depressed 
below,  while  the  south  rises  above  his  horizon,  and  continues  to  do  so  till 
the  south  pole  of  the  globe  is  reached,  when  that  of  the  heavens  will  be 
in  the  zenith. 

(121.)  The  same  stars,  in  their  diurnal  revolution,  come  to  the  meridian, 
sxtccemvely,  of  every  place  on  the  globe  once  in  twenty-four  sidereal  hours. 
And,  since  the  diurnal  rotation  is  uniform,  the  interval,  in  sidereal  time, 
which  elapses  between  the  same  star  coming  upon  the  meridians  of  two 
difierent  places  is  measured  by  the  difference  of  longitudes  of  the  places. 

(122.)  Vice  versd  —  the  interval  elapsing  between  two  different  stars 
coming  on  the  meridian  of  one  and  the  same  place,  expressed  in  sidereal 
time,  is  the  measure  of  the  difference  of  right  ascensions  of  the  stars. 

(123.)  The  equinoctial  intersects  the  horizon  in  the  east  and  west 
points,  and  the  meridian  is  a  point  whose  altitude  is  equal  to  the  co-lati- 
tude of  the  place.  Thus,  at  Greenwich,  of  which  the  latitude  is  51°  28' 
40",  the  altitude  of  the  intersection  of  the  equinoctial  and  meridian  is 
38"  81'  20".  The  north  and  south  poles  of  the  heavens  are  the  poles  of 
the  equinoctial.  The  east  and  west  points  of  the  horizon  of  a  spectator 
are  the  poles  of  his  celestial  meridian.  The  north  and  south  points  of  his 
horizon  are  the  poles  of  his  prime  vertical,  and  his  zenith  and  nadir  are 
the  poles  of  his  horizon. 

(124.)  All  the  heavenly  bodies  cidmmate  (i.  e.  come  to  their  greatest 
altitudes)  on  the  meridian ;  which  is,  therefore,  the  best  situation  to  ob- 
serve them,  being  least  confused  by  the  inequalities  and  vapours  of  the 
atmosphere,  as  well  as  least  displaced  by  refraction. 

(125.)  All  celestial  objects  within  the  circle  of  perpetual  apparition 
come  twice  on  the  meridian,  above  the  horizon,  in  every  diurnal  revolu- 
tion ;  once  above  and  once  heloio  the  pole.  These  are  called  their  icpper 
and  loicer  culminations. 

(126.)  The  problems  of  uranometry,  as  we  have  described  it,  consist 
in  the  solution  of  a  variety  of  spherical  triangles,  both  right  and  oblique 
angled,  according  to  the  rules,  and  by  the  formulae  of  spherical  trigononi- 
t'try,  which  we  suppose  known  to  the  reader,  oi-  for  which  he  will  consult 
appropriate  treatises.     "We  shall  only  here  observe  generally,  that  in  all 


1  \ 


74 


OUTLINES   OP  ASTRONOMY. 


problems  in  which  spherical  geometry  is  concerned,  the  student  will  find 
it  a  useful  practical  maxim  ratlior  to  consider  the  poles  of  the  great  circles 
which  the  question  before  him  rufers  to  than  the  circles  themselves.  To 
use,  for  example,  in  the  relations  he  has  to  consider,  polar  distances  rather 
than  declinations,  zenith  distances,  rather  than  altitudes,  &o,  Bearing 
this  in  mind,  there  are  few  problems  .'n  uninometry  which  will  offer  any 
difficulty.  The  followiug  are  the  crrabiuatious  which  most  commonly 
occur  for  solution  when  thejplace  of  one  celestial  object  only  on  the  sphere 
is  concerned.  ''-''  '■-'•    '■"■"'  '   '^'''  ■  ■---■  '■  ■   ^    ' " 

(127.)  In  the  triangle  ZPS,  Z  is  the  zenith,  T  the  elevated  pole,  and 
S  the  star,  sun,  or  other  celestial  objpct.  In  this  triangle  occur,  Ist,  PZ, 
which  being  the  complement  of  PH  (the  altitude  of  tha  pole),  is  ob- 
viously the  complement  of  the  latitude  (or  the  rolatihidp,  as  it  is  called) 
of  the  place ;  2d,  P  S,  iSie  polar  lUstance^  or  the  complement  of  the  decli- 
nation (co-decUnatlon)  of  the  star;  8d,  Z  8,  the  zenith  distance  or  co-altv- 
tudc  of  the  star.  If  P  S  bo  greater  than  90",  the  object  is  situated  on 
the  side  of  the  equinoctial  opposite  to  that  of  the  elevated  pole.  If  Z  S 
be  so,  the  object  is  below  the  horizon. 


In  the  same  triangle  the  angles  are,  Ist,  Z  P  S  the  lower  angle  j  2d,  PZ  S 
(the  supplement  of  S  Z  0,  which  latter  is  the  azimuth  of  the  star  or  other 
heavenly  body),  8d,  P  S  Z,  an  angle  which,  from  the  infrequency  of  any 
practical  reference  to  it,  has  nut  u(;quired  a  name.' 

'  In  the  practical  discussion  of  the  measures  of  double  stars  and  other  objects  by  the 
aid  of  the  position  micrometer,  this  angle  is  sometimes  required  to  be  known ;  and 
when  so  required,  it  will  be  not  niconveniently  referred  to  as  "  iho  arlglo  of  position 
ol  the  zenith." 


ELEMENTARY   KKL        jNS. 


75 


Tho  following  five  astronomical  magnitudes,  then,  occur  among  the  sides 
of  thia  most  useful  triangle  :  viz.,  Ist,  The  co-latitude  of  the  place  of 
observation;  2d,  the  polar  distance;  8d,  the  zenith  distance;  4th,  tho 
hour  angle;  and  5tb,  the  sub-azimuth  (supplement  of  azimuth)  of  a  given 
celestial  object;  and  by  its  solution  therefore  may  all  problems  bo  roHulvcd, 
in  which  three  of  these  magnitudes  are  directly  or  indirectly  given,  and 
the  other  two  required  to  be  found. 

(128.)  For  example,  suppose  the  time  of  rising  or  setting  of  the  sun 
or  of  a  star  were  required,  having  given  its  right  ascension  and  pohir  dis- 
tance. The  star  rises  when  apjiarentli/  on  the  horizon,  or  realli/  about 
34'  below  it  (owing  to  refraction),  so  that,  at  the  moment  of  its  ajiparent 
rising,  its  zenith  distance  is  90"  34'==Z  S.  Its  polar  listance  PS  ooiiig  Jilso 
given,  and  the  co-latitude  Z  P  of  the  place,  we  have  given  the  three  Midcs 
of  the  triangle,  to  find  the  hour  angle  Z  P  S,  which,  being  known,  is  to 
be  added  to  or  subtracted  from  the  star's  right  ascension,  to  give  the  side- 
real time  of  setting  or  rising,  which,  if  we  please,  may  be  converted  into 
solar  time  by  the  proper  rules  and  tables. 

(129.)  As  another  example  of  the  use  of  the  same  triangle,  we  may 
propose  to  find  the  local  sidereal  time,  and  the  latitude  of  the  place  of 
observation,  by  observing  equal  altitudes  of  the  same  star  east  and  west 
of  the  meridian,  and  noting  the  interval  of  the  observations  in  sidereal 
time.  ' 

The  hour  angles  corresponding  to  equal  altitades  of  a  fixed  star  being 
equal,  the  hour  angle  east  or  west  will  be  measured  by  half  the  observed 
interval  of  the  observations.  In  our  triangle,  then,  we  have  given  this 
hour  angle  Z  P  S,  the  polar  distance  P  S  of  the  star,  and  Z  S,  its  co- 
altitude  at  the  moment  of  observation.  Hence  we  may  find  P  Z,  tho 
co-latitude  of  the  place.  Moreover,  the  hour  angle  of  the  star  being 
known,  and  also  its  right  ascension,  the  point  of  the  equinoctial  is  known, 
which  is  on  the  meridian  at  the  moment  of  observation ;  and,  thr rcfore, 
the  local  sidereal  time  at  that  moment.  This  is  a  very  useful  observaitoii 
for  determining  the  latitude  and  time  at  an  unknown  station. 


76 


\ 


OUTLINES   OP  ASTRONOMY. 


CHAPTER  III.' 

OF  THE  NATURE  OP  ASTRONOMICAL  INSTRUMENTS  AND   OBSERVATIONS 

IN   GENERAL. 01  SIDEREAL   AND   SOLAR   TIME. — OF  THE  MEAWUKE- 

MENT8  OF  TIME.  —  CLOCKS,  CHRONOMETERS. — OF  ASTRONOMICAL 
MEASUREMENTS. — PRINCIPLE  OF  TELESCOPIC  SIGHTS  TO  INCREASE 
THE  ACCURACY  OF  POINTING.  —  SIMPLEST  APPLICATION  OF  THIS 
PRINCIPLE. —  THE  TRANSIT  INSTRUMENT. —  OF  THE  MEASUREMENT 
OF  ANGULAR  INTERVALS.  —  METHODS  OF  INCREASING  THE  ACCU- 
RACY OF  READING. —  THE  VERNIER. —  THE  MICROSCOPE. —  OF  THE 
MURAL  CIRCLE.  —  THE  MERIDIAN  CIRCLE. — FIXATION  OF  POLAR 
AND  HORIZONTAL  POINTS. — THE  LEVEL,  PLUMB-LINE,  ARTIFICIAL 
HORIZON. — PRINCIPLE  OF  COLLIMATION.— COLLIMATORS  OF  UITTEN- 
II0U8E,  KATER,  AND  BENZENBERO. — OF  COMPOUND  INSTRUMENTS 
WITH  CO-ORDINATE  CIRCLES. — THE  EQUATORIAL,  ALTITUDE,  AND 
AZIMUTH  INSTRUMENTS.  —  THEODOLITE.  —  OF  THE  SEXTANT  AND 
REFLECTING  CIRCLE. — PRINCIPLE  OF  REPETITION.  —  OF  MICROME- 
TERS.—  PARALLEL  WIRE  MICROMETER. — PRINCIPLE  OF  THE  DU- 
PLICATION OF  IMAGES. — THE  IIELIOMETER. —  DOUBLE  REFRACTING 
EYE-PIECE. — VARIABLE  PRISM  MICROMETER. — OF  THE  POSITION 
MICROMETER. 


(130.)  Our  first  chapters  have  been  devoted  to  the  acquisition  chiefly 
of  preliminary  notions  respecting  the  globe  we  inhabit,  its  relation  to  the 
celestial  objects  which  surround  it,  and  the  physical  circumstances  under 
which  all  astronomical  observitions  must  be  made,  as  well  as  to  provide 
ourselves  with  a  stock  of  tecinical  words  and  elementary  ideas  of  most 
frequent  and  familiar  use  in  the  sequel.  We  might  now  proceed  to  a 
more  exact  and  detailed  statement  of  the  facts  and  theories  of  astronomy ; 
but,  in  order  to  do  this  with  full  effect,  it  will  be  desirable  that  the 
reader  be  made  acquainted  with  the  principal  means  which  astronomers 

'  The  student  who  is  anxious  to  become  acquainted  with  the  chief  subject  matter 
of  this  work,  may  defer  the  reading  of  that  part  of  this  chapter  which  is  devoted  to 
the  description  of  particular  instruments,  or  content  himself  with  a  cursory  perusal 
of  It,  until  farther  advanced,  when  it  will  be  necessary  to  return  to  it. 


NATURE  OP   ASTRONOMICAL   INSTRUMENTS. 


77 


posscBS,  of  detorrainlng,  with  tho  dogreo  of  nicety  their  theories  require, 
the  duta  on  which  they  ground  their  conclusions ;  in  other  word.s,  of  as- 
certaining by  measurement  the  apparent  and  real  magnitudes  with  which 
thoy  are  conversant.  It  is  only  when  in  possession  of  this  knowledge 
that  he  can  fully  appreciate  cither  the  truth  of  tho  theories  themselves, 
or  tho  degree  of  reliance  to  be  placed  on  any  of  their  conclusions  ante- 
cedent to  trial :  since  it  is  only  by  knowing  what  amount  of  error  can 
certainly  be  perceived  and  distinctly  measured,  that  he  can  satisfy  himself 
whether  any  theory  offers  so  close  an  approximation,  in  its  numerical 
results,  to  actual  phenomena,  as  will  justify  him  in  receiving  it  as  a  true 
representation  of  nature. 

(131.)  Astronomical  instrument-making  may  be  justly  regarded  as  tho 
most  refined  of  the  mechanical  arts,  and  that  in  which  the  nearest  ap- 
proach to  geometrical  precision  is  required,  and  has  been  attained.     It 
may  be  thought  an  easy  thing,  by  one  unacquainted  with  tho  niceties  re- 
quired, to  turn  a  circle  in  metal,  to  divide  its  circumference  into  3G0 
equal  parts,  and  these  again  into  smaller  subdivisions,  —  to  place  it  accu- 
rately on  its  centre,  and  to  adjust  it  in  a  given  position ;  but  practically 
it  is  found  to  be  one  of  tho  most  difficult.     Nor  will  this  appear  extra- 
ordinary, when  it  is  considered  that,  owing  to  the  application  of  telescopes 
to  tho  purposes  of  angular  measurement,  every  imperfection  of  structure 
of  division  becomes  magnified  by  the  whole  optical  power  of  that  instru- 
ment ;  and  that  thus,  not  only  direct  errors  of  workmanship,  arising  from 
unsteadiness  of  hand  or  imperfection  of  tools,  but  those  inaccuracies 
which  originate  in  far  more  uncontrollable  causes,  such  as  the  unequal 
expansion  and  contraction  of  metallic  masses,  by  a  change  of  temperature, 
and  their  unavoidable  flexure  or  bending  by  their  own  weight,  become 
perceptible  and  measurable.     An  angle  of  one  minute  occupies,  on  the 
circumference  of  a  circle  of  10  inches  in  radius,  only  about  -jjuth  part 
of  an  inch,  a  quantity  too  small  to  be  certainly  dealt  with  without  the 
use  of  magnifying  glasses;   yet  one  minute  is  a  gross  quantity  in  the 
astronomical  measurement  of  an  angle.     With  the  instruments  now  em- 
ployed in  observatories,  a  single  second,  or  the  60th  part  of  a  minute,  is 
rendered  a  distinctly  visible  and  appreciable  quantity.     Now,  the  arc  of 
a  circle,  subtended  by  one  second,  is  less  than  the  •200,000th  part  of  the 
radius,  so  that  on  a  circle  of  6  feet  in  diameter  it  would  occupy  no  greater 
linear  extent  than  37  jj^th  part  of  an  inch ;  a  quantity  requiring  a  power- 
ful microscope  to  be  discerned  at  all.     Let  any  one  figure  to  himself, 
therefore,  the  difficulty  of  placing  on  the  circumference  of  a  metallic 
circle  of  such  dimensions  (supposing  the  difficulty  of  its  construction  sur- 
mounted), 360  marks,  dots,  or  cognizable  divisions,  which  shall  all  be 


78 


OUTLINES   OF  ASTRONOMY. 


true  to  their  places  within  such  narrow  limits ;  to  say  nothing  of  the  sub- 
division  of  the  degrees  so  marked  off  into  minutes,  and  of  these  again 
into  seconds.  Such  a  work  has  probably  baffled,  and  will  probably  for 
ever  continue  to  baffle,  the  utmost  stretch  of  human  skill  and  industry ; 
nor,  if  executed,  could  it  endure.  The  ever-varying  fluctuations  of  heat 
and  cold  have  a  tendency  to  produce  not  merely  temporary  and  transient, 
but  pemanent,  uncompensated  changes  of  form  in  all  considerable  masses 
of  those  metals  which  alone  are  applicable  to  such  uses ;  and  their  own 
weight,  however  symmetrically  formed,  must  always  be  unequally  sus- 
tained, since  it  is  impossible  to  apply  the  sustaining  power  to  every  part 
separately :  even  could  this  be  done,  at  all  events  force  must  be  used  to 
move  and  to  fix  them ;  which  can  never  be  done  without  producing  tem- 
porary and  risking  permanent  change  of  form.  It  is  true,  by  dividing 
them  on  their  centres,  and  in  the  identical  places  they  are  destined  to 
occupy,  and  by  a  thousand  ingenious  and  delicate  contrivances,  wonders 
have  been  accomplished  in  this  department  of  art,  and  a  degree  of  perfec- 
tion has  been  given,  not  merely  to  chefs  d'ceuvre,  but  to  instruments  of 
moderate  prices  and  dimensions,  and  in  ordinary  us?,  which,  on  due  con- 
sideration, must  appear  very  surprising.  But  though  we  are  entitled  to 
look  for  wonders  at  the  hands  of  scientific  artists,  we  are  not  to  expect 
miracles.  The  demands  of  the  astronomer  will  always  surpass  the  power 
of  the  artist ;  and  it  must,  therefore,  be  constantly  the  aim  of  the  former 
to  make  himself,  as  far  as  possible,  independent  of  the  imperfections  inci- 
dent to  every  work  the  latter  can  place  in  his  hands.  He  must,  therefore, 
endeavour  so  to  combine  his  observations,  so  to  choose  his  opportunities, 
and  so  to  familiarize  himself  with  all  the  causes  which  may  produce  in- 
strumental demngcnient,  and  with  all  the  peculiarities  of  structure  and 
material  of  each  instrument  he  possesses,  as  not  to  allow  himself  to  be 
misled  by  their  errors,  but  to  extract  from  their  indications,  as  far  as  pos- 
sible, all  that  is  true,  and  reject  all  that  is  erroneous.  It  is  in  this  that 
the  art  of  the  practical  astronomer  consists, — an  art  of  itself  of  a  curious 
and  intricate  nature,  and  of  which  we  can  here  only  notice  some  of  the 
leading  and  general  features. 

(132.)  The  great  aim  of  the  practical  astronomer  being  numerical 
correctness  in  the  results  of  instrumental  measurement,  his  constant  care 
and  vigilance  must  be  directed  to  the  detoct'on  and  compensation  of  errors, 
cither  by  annihilating,  or  by  taking  account  of,  and  allowing  for  them. 
Now,  if  we  examine  the  sources  from  which  errors  may  arite  in  any 
instrumental  determination,  we  shall  find  them  chiefly  reducible  to  three 
principal  heads :  —  i 

(183.)  1st,  External  or  incidental  causes  of  error;  coxoprehending 


INSTRUMENTAL  AND   OTHER   SOURCES   OF  ERROR. 


79 


such  as  depend  on  external,  uncontrollable  circumstpances :  such  as,  fluc- 
tuations of  weather,  which  dijturb  the  amount  of  refraction  from  its  tabu- 
lated value,  and,  being  reducible  to  no  fixed  law,  induce  uncertainty  to 
the  extent  of  their  own  possible  magnitude  j  such  as,  by  varying  the  tem- 
perature of  the  air,  vary  also  the  form  and  position  of  the  instruments 
used,  by  altering  the  relative  magnitudes  and  the  tension  of  their  parts ; 
and  others  of  the  like  nature.  ,     . 

(134.)  2dly,  Errors  of  observation :  such  as  arise,  for  example,  from 
mexpertness,  defective  vision,  slowness  in  sf^izing  the  exact  instant  of 
occurrence  of  a  phenomenon,  or  precipitancy  in  anticipating  it,  &c. ; 
from  atmospheric  indistinctness ;  insufficient  optical  power  in  the  instru- 
ment, and  the  like.  Under  this  head  may  also  be  classed  all  errors  arising 
from  momentary  instrumental  derangement, — slips  in  clamping,  looseness 
of  screws,  &c. 

(135.)  3dly,  The  third,  and  by  far  the  most  numerous  class  of  errors 
to  which  astronomical  measurements  are  liable,  arise  from  causes  which 
may  be  deemed  instrumental,  and  which  may  be  subdivided  into  two 
principal  classes.  The  ^rs<  comprehends  those  which  arise  from  an  instru- 
ment not  heing  what  it  professes  to  be,  which  is  error  of  workmanship. 
Thus,  if  a  pivot  or  axis,  instead  of  being,  as  it  ought,  exactly  cylindrical, 
be  slightly  flattened,  or  elliptical, — if  it  be  not  exactly  (as  it  is  intended 
it  should)  concentric  with  the  circle  it  carries;— if  this  circle  (so  called) 
be  in  reality  not  exactly  circular,  or  not  in  one  p'^ne;  —  if  its  divisions, 
intended  to  be  precisely  equidistant,  should  be  placed  in  reality  at  unequal 
intervals, — and  a  hundred  other  things  of  the  same  sort.  These  are  not 
mere  speculative  sources  of  error,  but  practical  annoyances,  which  every 
observer  has  to  contend  with. 

(130.)  The  other  subdivision  of  instrumental  errors  comprehends  such 
as  arise  from  au  instrument  not  being  placed  in  the  position  it  ought  to 
have ;  and  from  those  of  its  parts,  which  are  made  purposely  moveable, 
not  being  properly  disposed  inter  se.  These  are  errors  of  adjustment. 
Some  are  unavoidable,  as  they  arise  from  a  general  unsteadiness  of  the 
soil  or  building  in  which  the  instruments  are  placed  j  which,  though  too 
minute  to  be  noticed  in  any  other  way,  become  appreciable  in  delicate 
astronomical  observations;  others,  again,  are  consequences  of  imperfect 
workmanship,  as  where  an  instrument  once  well  adjusted  will  not  remain 
so,  but  keeps  deviating  and  shifting.  But  the  most  important  of  this  class 
of  errors  arise  from  the  non-existence  of  natural  indications,  other  than 
those  afforded  by  astronomical  observations  themselves,  whether  an  instru 
ment  has  or  has  not  the  exact  position,  with  respect  to  the  horizon  and  ita 


■"'^r 


--■"■tjy  ~j-«H*ijrs^*fV'i-rf'^f)T'™i;™»F.j|7KT'  ■ 


^0 


OUTLINES   OF  ASTRONOMY. 


cardinal  points,  the  axis  of  the  earth,  or  to  other  principal  astronomical 
lines  and  circles,  which  it  ought  to  have  to  fulfil  properly  its  objects. 

(137.)  Now,  with  respect  to  the  first  two  classes  of  error,  it  must  be 
observed,  that,  in  so  far  as  they  cannot  be  reduced  to  known  laws,  and 
thereby  become  subjects  of  calculation  and  due  allowance,  they  actually 
vitiate,  to  their  full  extent,  the  results  of  any  observations  in  which  they 
subsist.  Being,  however,  in  their  nature  casual  and  accidental,  their 
effects  necessarily  lie  sometimes  one  way,  sometimes  the  other;  sometimes 
diminishing,  sometimes  tending  to  increase  the  results.  Hence,  by  greatly 
multiplying  observations,  under  varied  circumstances,  by  avoiding  unfa- 
vourable, and  taking  advantage  of  favourable  circumstances  of  weather, 
or  otherwise  using  opportunity  to  advantage  —  and  finally,  by  taking  the 
meaji  or  average  of  the  results  obtained,  this  class  of  errors  may  be  so 
far  subdued,  by  setting  them  to  destroy  one  another,  as  no  longer  sensibly 
to  vitiate  any  theoretical  or  practical  conclusion.  This  is  the  great  and 
indeed  only  resource  against  such  errors,  not  merely  to  the  astronomer, 
but  to  the  investigator  of  numerical  results  in  every  department  of 
physical  research. 

(138.)  With  regard  to  errors  of  adjustment  and  workmanship,  not 
only  the  po.s.«'i<7/Vy,  but  the  certainty  of  their  existence,  in  every  ima- 
ginable form,  ic  all  instruments,  must  be  contemplated.  Human  hands 
or  machines  nc  er  formed  a  circle,  drew  a  straight  line,  or  erected  a  per- 
pendicular, nor  ever  placed  an  instrument  in  i^erfect  adjustment,  unless 
accidentally ;  and  then  only  during  an  instant  of  time.  This  does  not 
prevent,  however,  that  a  great  approximation  to  all  these  desiderata 
should  be  attained.  But  it  is  the  peculiarity  of  astronomical  observation 
to  be  the  ultimate  means  of  detection  of  all  mechanical  defects  which 
elude  by  their  minuteness  every  other  mode  of  detection.  What  the  eye 
cannot  discern  nor  the  touch  perceive,  a  course  of  astronomical  observa- 
tions will  make  distinctly  evident.  The  imperfect  products  of  man's 
hands  are  here  tested  by  being  brouglit  into  comparison  under  very  great 
magnifying  power?  (corresponding  in  effect  to  a  great  increase  in  acute- 
ness  of  perception)  with  the  perfect  workmanship  of  nature;  and  there 
is  none  which  will  bear  the  trial.  Now,  it  may  seem  like  arguing  in  a 
vicious  circle,  to  deduce  theoretical  conclusions  and  laws  from  observation, 
and  then  to  turn  I'ound  upon  the  instruments  with  which  those  observa- 
tions were  made,  accuse  them  of  imperfection,  and  attempt  to  detect  and 
rectify  their  '^nors  by  means  of  the  very  laws  and  theories  which  they 
have  helped  ua  to  a  knowledge  of  A  little  consideration,  however,  will 
suffice  to  show  that  such  a  course  of  proceeding  is  perfectly  legitimate. 

(139.)  The  steps  by  which  we  arrive  at  the  laws  of  natural  phenomena, 


1 


''JTfJ-'T?*;'??!™'"'' 


INSTRUMENTAL   SOURCES   OF   ERROR. 


81 


and  especially  those  which  depend  for  their  verification  on  numerical 
determinations,  are  necessarily  successive.  Gross  results  and  palpable 
laws  are  arrived  at  by  rude  observation  with  coarse  instruments,  or 
without  any  instruments  at  all,  and  are  expressed  in  language  which  is 
not  to  be  considered  as  absolute,  but  is  to  be  interpreted  with  a  degree  of 
latitude  commensurate  to  the  imperfection  of  the  observations  themselves. 
These  results  are  corrected  and  refined  by  nicer  scrutiny,  and  with  more 
delicate  means.  The  first  rude  expressions  of  the  laws  which  embody 
them  are  perceived  to  be  inexact.  The  language  used  in  their  expression 
is  corrected,  its  terms  more  rigidly  defined,  or  fresh  terms  introduced, 
until  the  new  state  of  language  and  terminology  is  brought  to  fit  the 
improved  state  of  knowledge  of  facts.  In  the  progress  of  this  scrutiny 
subordinate  laws  are  brought  into  view  which  still  further  modify  both 
the  verbal  statement  and  mTmerical  results  of  those  which  first  offered 
themselves  to  our  notice ;  and  when  these  are  traced  out  and  reduced  to 
certainty,  others,  again,  subordinate  to  them,  make  their  appearance,  and 
become  subjects  of  further  inquiry.  Now,  it  invariably  happens  (and 
the  reason  is  evident)  that  the  first  glimpse  we  catch  of  such  subordinate 
laws  —  the  first  form  in  which  they  are  dimly  shadowed  out  to  our  minds 
— is  that  of  errors.  "We  perceive  a  discordance  between  what  we  expect, 
and  what  we  Jind.  The  first  occurrence  of  such  a  discordance  we  attri- 
bute to  accident.  It  happens  again  and  again ;  and  we  begin  to  suspect 
our  instruments.  We  then  inquire,  to  what  amount  of  error  their  deter- 
minations can,  hi/  possihility,  be  liable.  If  their  limit  of  possible  error 
exceed  the  observed  deviation,  we  at  once  condemn  the  instrument,  and 
set  about  improving  its  construction  or  adjustments.  Stii'  he  same 
deviations  occur,  and,  so  far  from  being  palliated,  are  more  raarked  and 
better  defined  than  before.  We  are  now  sure  that  we  are  on  the  traces 
of  a  law  of  nature,  and  we  pursue  it  till  we  have  reduced  it  to  :i  definite 
ftiitoment,  and  verified  it  by  repeated  observation,  uuJvir  every  variety 
of  circumstances. 

(140.)  Now,  in  the  course  of  this  inquiry,  it  will  not  fail  to  happen 
that  other  discordances  will  strike  us.  Taught  by  experience,  we  suspect 
the  existence  of  some  natural  law,  before  unknown ;  we  tabulate  (i.  e. 
draw  out  in  order)  the  results  of  our  observations ;  and  we  perceive,  in 
ibis  synoptic  statement  of  them,  distinct  indications  of  a  regular  progres- 
sion. Again  we  improve  or  vary  our  instruments,  and  we  now  lose  sight 
of  this  supposed  new  law  of  nature  altogether,  or  find  it  replaced  by  some 
other,  of  a  totally  different  character.  Thus  we  are  led  to  suspect  an 
instrumental  cause  for  what  we  have  noticed.  We  examine,  therefore, 
the  theory  of  our  instrument ;  we  suppose  defects  in  its  structure,  and,  by 
6 


OUTLINES   OF  ASTRONOMY. 


the  aid  of  geometry,  we  trace  their  influence  in  introducing  arfudl «"'  ^ra 
into  its  indications.  These  errors  have  their  latvs,  which,  so  long  ad  we 
have  no  knowledge  of  causes  to  guide  us,  may  be  confounded  with  laws 
of  nature,  as  they  are  mixed  up  with  them  in  their  effects.  They  are  not 
fortuitous,  like  errors  of  observation,  but,  as  they  arise  from  sources 
inherent  in  the  instrument,  and  unchangeable  while  it  and  its  adjustments 
remain  unchanged,  they  are  reducible  to  fixed  and  ascertainable  forms ; 
each  particular  defect,  whether  of  structure  or  adjustment,  producing  its 
own  appropriate  form  of  error.  V/hen  these  are  thoroughly  investigated, 
we  recognize  among  them  one  which  coincides  in  its  nature  and  progression 
with  that  of  our  observed  discordances.  The  mystery  is  at  once  solved. 
We  have  detected,  by  direct  observation,  an  instrumental  defect. 

(141.)  It  is,  therefore,  a  chief  requisite  for  the  practical  astronomer  to 
make  himself  completely  familiar  with  the  tlieory  of  his  instruments.  By 
this  alone  is  he  enabled  at  once  to  decide  what  effect  on  his  observations  any 
given  imperfection  of  structure  or  adjustment  will  produce  in  any  given 
circumstances  under  which  an  observation  can  be  made.  This  alone  also 
can  place  him  in  a  condition  to  derive  available  and  practical  means  of 
destroying  and  eliminating  altogether  the  influence  of  such  imperfections, 
by  so  arranging  his  observations,  that  it  shall  affect  their  results  in  oppo- 
site ways,  and  that  its  influence  shall  thus  disappear  from  their  mean, 
which  is  one  of  the  chief  modes  by  which  precision  is  attained  in  practical 
astronomy.  Suppose,  for  example,  the  principle  of  an  instrument  required 
that  a  circle  should  be  concentric  with  the  axis  on  which  it  is  made  to 
turn.  As  this  is  a  condition  which  no  workmanship  can  exactly  fulfil,  it 
becomes  necessary  to  inquire  what  errors  will  be  produced  in  observations 
made  and  registered  on  the  faith  of  such  an  instrument,  by  any  assigned 
deviation  in  this  respect;  that  is  to  say,  what  would  be  the  disagreement 
between  observations  made  with  it  and  with  one  absolutely  perfect,  could 
such  be  obtained.  Now,  simple  geometrical  conside rations  suffice  to  show 
—  1st.  that  if  the  axis  be  excentric  by  a  given  fraction  (say  one  thou- 
sandth part)  of  the  radius  of  the  circle,  all  angles  re;vd  off"  on  that  part 
of  the  circle  towards  which  the  excf^ntricity  lies,  will  ap}y<^arby  that  frac- 
tional amount  too  smal',  and  all  oi  the  opposite  side  too  large.  And, 
2dly,  fuat  tchaU  er  be  tl:e  amount  of  the  excentricity,  and  on  whatever 
part  of  the  circl  any  proposed  anglo  is  measured,  the  effect  of  the  error 
in  question  on  the  result  of  observations  depending  on  tb#  graduation  of 
its  circumference  (or  limb,  as  it  is  technically  called)  wilj  b*  con»pl'  wly 
annihilated  by  the  very  easy  method  of  alway*-  reading  o#  cie  divisions 
on  two  diametrically  opposite  points  >f  th«  circle,  and  taking  a  mean  ,  far 
the  effect  of  excentricity  is  always  Ut  Mereaae  f^m  arc  rvfr*tw^xxi\u^  Sitt 


ILLUSTRATION   BY   EXAMPLES.  —  REFRACTION.  ill 

angle  in  question  on  one  side  of  the  circle,  i^y  just  the  same  quantity  by 
which  it  diminii^hes  that  on  the  other.  Again,  suppose  that  the  proper 
use  of  the  instrument  required  that  this  axis  should  be  exactly  parallel  to 
that  of  the  earth.  As  it  never  can  he  placed  or  remain  so,  it  becorriv-s  a 
question,  what  amount  of  error  will  arise,  in  its  use,  from  any  ji3sif;;Qed 
deviation,  whether  in  a  horizontal  or  vertical  plane,  from  this  precise  posi- 
tion. Such  inquiries  constitute  the  theory  of  instrumental  errors;  a 
theory  of  the  utmost  importance  to  practice,  and  one  of  which  a  complete 
knowledge  will  enable  an  observer,  with  moderate  instrumental  means, 
often  to  attain  a  degree  of  precision  which  might  seem  to  belong  only  to 
the  most  refined  and  costly.  This  theory,  as  will  readily  be  apprehended, 
turns  almost  entirely  on  considerations  of  pure  geometry,  and  those  for 
the  most  part  not  difGcult.  In  the  present  work,  however,  we  have  no 
further  concern  with  it.  The  Astronomical  instruments  we  propose  briefly 
to  describe  in  this  chapter  will  be  considered  as  perfect  both  in  construe- 
tioD  and  adjustment.' 

(142.)  As  the  above  remarks  are  very  essential  to  a  right  understari- 
ing  of  the  philosophy  of  our  subject  and  the  spirit  of  astronomical 
methods,  we  shall  elucidate  them  by  taking  one  or  two  special  cases. 
Observant  persons,  before  the  invention  of  astronomical  instruments,  had 
already  concluded  the  apparent  diurnal  motions  of  the  stars  to  be  per- 
formed in  circles  about  fixed  poles  in  the  heavens,  as  shown  in  the  fore- 
going chapter.  In  drawiug  this  conclusion,  however,  refraction  was 
entirely  overlooked,  or,  if  forced  oa  their  notice  by  its  great  magnitude 
in  the  immediate  neighbourhood  of  the  horizon,  was  regarded  as  a  local 
irregularity,  and,  as  such,  neglected,  or  slurred  over.  As  soon,  however, 
as  the  diurnal  paths  of  the  stars  were  attempted  to  be  traced  by  instru- 
ments, even  of  the  coarsest  kind,  it  became  evident  that  the  notion  of 
exact  circles  described  about  one  and  the  same  pole  would  not  represent 
the  phenomena  correctly,  but  that,  owing  to  some  cause  or  other,  the 
apparent  diurnal  orbit  of  every  star  is  distorted  from  a  circular  into  an 
oval  form,  its  lower  segment  being  flatter  than  its  upper ;  and  the  devia- 
tion being  greater  the  nearer  the  star  approached  the  horizon,  the  effect 
being  the  same  as  if  the  circle  had  been  squeezed  upwrrds  from  below, 
and  the  lower  parts  more  than  the  higher.  For  such  an  effect,  as  it  was 
soon  found  to  arise  from  no  casual  or  instrumental  cause,  it  became  neces- 
sary to  seek  a  natural  one  j  and  refraction  readily  occurred,  to  solve  the 


'  The  principle  on  which  the  chief  adjustments  of  two  or  three  of  the  mo8t  useful 
and  common  instruments,  such  as  the  transit,  the  equatorial,  and  the  sextant,  are  per 
fo! (iifd,  are,  however,  noticed,  for  the  convenience  of  readers  who  may  use  sucn  hi 
strumcnts  without  going  farther  into  the  arcana  of  practical  astronomy. 


84 


OLTLIXES   OF  ASTRONOMT. 


difficulty.  In  fact,  it  is  a  case  precisely  analogous  to  what  we  have 
already  noticed  (art.  47),  of  the  apparent  distortion  of  the  sun  near  the 
horizon,  only  on  a  larger  scale,  and  traced  up  to  greater  altitudes.  This 
new  law  once  established,  it  became  necessary  to  modify  the  expression  of 
that  anciently  received,  by  inserting  in  it  a  salvo  for  the  effect  of  refrac- 
tion, or  by  making  a  distinction  between  the  a/pparent  diurnal  orbits,  as 
affected  by  refraction,  and  the  true  ones  cleared  of  that  effect.  This  dis- 
tinction between  the  apparent  and  the  true — between  the  uncorrected 
and  corrected — between  the  rotigh  and  obviouSf  and  the  refined  and  ulti- 
mate— is  of  perpetual  occurrence  it,  every  part  of  astronomy. 

(I'vy  \gain.  The  first  impression  produced  by  a  view  of 'he  diurnal 
movement  of  the  heavens  is  that  all  the  heavenly  bodies  perform  this 
revf  lusii  a  in  one  common  period,  viz.  a  day,  or  24  hours.  But  no  sooner 
do  we  come  to  examine  the  matter  instrumeniallf/f  i.  e.  by  noting,  by 
rrjO' keepers,  their  successive  arrivals  on  the  meridian,  than  we  find  dif- 
i-  i-ences  which  cannot  be  accounted  for  by  any  error  of  observation.  All 
thv-  f.  ,  16  is  true,  occupy  the  same  interval  of  time  between  their  suc- 
cessive appulses  to  the  meridian,  or  to  any  vertical  circle ;  but  this  in  a 
very  different  one  from  that  occupied  by  the  sun.  It  is  palpably  shorter ; 
being,  in  fact,  only  23'>  56'  409",  instead  of  24  houra,  such  hours  as 
our  common  clocks  mark.  Here,  t'>3n,  we  have  already  two  different 
days,  a  sidereal  and  a  solar;  and  if,  instead  of  the  sun,  we  observe  the 
moon,  we  find  a  third,  much  longer  than  either, — a  lunar  day,  whose 
average  duration  is  24*  54"  of  our  ordinary  time,  which  last  is  solar  time, 
being  of  necessity  conformable  to  ihe  sun's  successive  re-appearances,  on 
which  all  the  business  of  life  depends. 

(144.)  Now,  all  the  stars  are  found  to  bo  unanimous  in  giving  the 
same  exact  duration  of  23"  56'  4-09  ",  for  the  sidereal  day;  which,  there- 
fore, we  cannot  hesitate  to  receive  as  the  period  in  which  the  earth  makes 
one  revolution  on  its  axis.  We  are,  therefore,  compelled  to  look  on  the 
sun  and  moon  as  exceptions  to  the  funeral  law ;  as  having  a  different 
nature,  or  at  least  a  different  relation  to  us,  from  thfi  stars ;  and  as  having 
motions,  real  or  apparent,  of  their  own,  independ  at  of  the  rotation  of 
the  earth  on  its  axis.  Thus  a  great  and  most  important  diHtinction  is 
disclosed  to  us. 

(145.)  To  establish  these  facts,  almost  no  apparatus  is  required.  An 
observer  need  only  station  himself  to  the  north  of  some  well-defined  ver- 
tical object,  as  the  angle  of  a  building,  and,  placing  bis  eye  exactly  at  a 
certain  fixed  point  (such  as  a  small  hol.^  iu  a  plate  of  metal  ;  tilod  to  some 
immovei  ble  support),  notice  the  successive  dijsuppearauces  of  any  star  be- 


m\ 


SIDEREAL  AND   SOLAR  TIME. 


85 


bind  the  building,  by  a  watch.'  When  he  observes  the  sun,  he  must 
shade  his  eye  with  a  dark-coloured  or  smoked  glass,  and  notice  the  moments 
when  its  western  and  eastern  edges  successively  come  up  to  the  wall,  from 
which,  by  taking  half  the  interval,  he  will  ascertain  (what  he  cannot  di- 
rectly observe)  the  moment  of  disappearance  of  its  centre.       *-     > 

(146.)  When,  in  pursuing  and  establishing  this  general  fact,  we  are 
led  to  attend  more  nicely  to  the  times  of  the  daily  arrival  of  the  sun  on 
the  meridian,  irregularities  (such  they  first  seem  to  be)  begin  to  make  their 
appearance.  The  intervals  between  two  successive  arrivals  are  not  the  same 
at  all  times  of  the  year.  They  are  sometimes  greater,  sometimes  less,  than 
24  hours,  as  shown  by  the  clock ;  that  is  to  say,  the  solar  day  is  not  always 
of  the  same  length.  About  the  21st  of  December,  for  example,  it  is  half 
a  minute  longer,  and  about  the  same  day  of  September  nearly  as  much 
shorter,  than  its  average  duration.  And  thus  a  distinction  is  again  press- 
ed upon  our  notice  between  the  actual  soUr  day,  which  is  never  two  days 
in  succession  alike,  and  the  mean  solar  day  of  24  hours,  which  is  an  ave- 
rage of  all  the  solar  days  throughout  the  year.  Here,  then,  a  new  source 
of  inquiry  opens  to  us.  The  sun's  apparent  motion  is  not  only  not  the 
same  with  the  stars,  but  it  is  not  (as  the  latter  is)  uniform.  It  is  subject 
to  fluctuations,  whose  laws  become  matter  of  investigation.  But  to  pur- 
sue these  laws,  we  require  nicer  means  of  observation  than  what  we  have 
described,  and  are  obliged  to  call  in  to  our  aid  an  instrument  called  the 
transit  instrument,  especially  destined  for  such  observations,  and  to  attend 
minutely  to  all  the  causes  of  irregularity  in  the  going  of  clocks  and  watches 
which  may  affect  our  reckoning  of  time.  Thus  we  become  involved  by 
degrees  in  iiiore  and  more  deli  .mj  instniraont*.!  inquiries;  and  we  speed- 
ily find  that,  in  proportion  as  we  aseei'tain  the  au.ount  and  law  of  one 
great  or  leading  fluctuation,  or  inequality,  as  it  is  called,  of  the  sun's 
diurnal  motion,  we  bring  into  view  others  continually  smaller  and  smaller, 
which  were  before  obscured,  or  mixed  up  with  errors  of  observation  and 
instrumeulal  imperfections.  In  short,  we  may  not  inaptly  compare  the 
mean  length  of  the  solar  day  to  the  mean  or  average  height  of  water  in 
a  harbour,  or  the  genera!  level  of  the  sea  unagitated  by  tide  or  waves. 
The  great  annual  fluctuation  aWve  noticed  may  be  compared  to  the  daily 

'  This  is  nn  excellent  ^actioal  method  of  ascertaining  the  raie  of  a  clock  or  watch, 
being  ^xceeimg  accurate  if  a  few  precautions  are  attended  to;  the  chief  of  which  is, 
to  lake  care  that  tliat  part  of  the  effge  behind  which  the  star  (a  bright  one,  7iot  a  planet) 
disappears  shall  lie  quiie  smooth  ;  as  otherwise  variable  refraction  may  transfer  tho 
point  of  disappearance  from  a  protuberance  to  a  notch,  and  thus  vary  tho  moment  of 
observation  undulv.  This  is  easily  sccur.,^,  *>y  nailing  up  a  sicooth-edged  hoara. 
The  verticality  of  us  edge  should  be  insured  by  the  use  of  a  plumb-line. 


^■^ 


8ti 


OUTLINES   OP   ASTRONOMY. 


variationa  of  level  produced  by  the  tides,  which  are  nothing  but  enormoua 
waves  extending  over  the  whole  ocean,  while  the  smaller  subordinate  ine- 
qualities may  be  assimilated  to  waves  ordinarily  so  called,  on  which,  when 
large,  we  perceive  lesser  undulations  to  ride,  and  on  these,  again,  minuter 
ripplings,  to  the  series  of  whose  subordination  we  can  perceive  no  end. 

(147.)  With  the  causes  of  these  irregularities  in  the  solar  motion  we 
have  no  concern  at  present ;  their  explanation  belongs  to  a  more  advanced 
part  of  our  subject;  but  the  distinction  between  the  solar  and  sidereal 
days,  as  it  pervades  every  part  of  astronomy,  requires  to  be  early  intro- 
duced, and  never  lost  sight  of.  It  is,  as  already  observed,  the  mean  or 
average  length  of  the  solar  day,  which  ia  used  in  the  civil  reckoning  of 
time.  It  commences  at  midnight,  but  astronomers,  even  when  they  use 
mean  solar  time,  depart  from  the  civil  reckoning,  commencing  their  day 
at  noon,  and  reckoning  the  hours  from  0  ro\ind  to  24.  Thus,  11  o'clock 
in  the  forenoon  of  the  second  of  January,  in  the  civil  reckon' ng  of  time, 
corresponds  to  January  1  day  23  hours  in  the  astronomical  reckoning ; 
and  one  o'clock  in  the  afternoon  of  the  former,  to  January  2  days  1  hour 
of  the  latter  reckoning.  This  usage  has  its  advantages  and  disadvantages, 
but  the  latter  seem  to  preponderate ;  and  it  would  be  well  if,  in  conse- 
quence, it  could  be  broken  through,  and  the  civil  reckoniag  substituted. 
Uniformity  in  nomenclature  and  modes  of  reckoning  in  all  matters  relat- 
ing to  time,  space  J  weight,  measure,  &c.,  is  of  such  vast  and  paramount 
importance  in  every  relation  of  life  as  to  outweigh  every  consideration  of 
technical  convenLnce  or  custom.* 

(148.)  Both  astronomers  and  civilians,  however,  who  inhabit  different 
points  of  the  earth's  surface,  differ  from  each  other  in  their  reckoning  of 
time ;  ns  it  is  obvious  they  must,  if  we  coiidider  that,  when  it  is  noon  at 
one  place,  it  is  midnight  at  a  place  diametrically  opposite;  sunrise  at 
anfither;  and  sunset^  again,  at  a  fourth.  Hence  arises  considerable  in- 
cocvenience,  especially  as  respects  places  differing  very  widely  in  situation, 
and  which  may  even  in  some  critical  cases  involve  the  mistake  of  a  whole 
day .  To  obviate  this  inconvenience,  there  has  lately  been  introduced  a 
system  of  reckoning  time  by  mean  solar  days  and  parts  of  a  day  counted 
from  a  fixed  instant,  common  to  all  the  world,  and  determined  by  no  local 

'  The  only  disadvantage  to  ofitrcmomers  of  using  the  civi!  reckoning  is  this — that 
tbrir  observations  being  chiefly  carried  on  during  the  night,  the  day  of  their  date  will, 
in  this  reckoning,  always  have  to  be  changed  at  midnight,  and  the  former  and  latter 
por'ion  of  every  night's  observations  will  belong  to  two  difTereiitiy  numbered  civil  days 
111  the  month.  There  is  no  denying  this  to  be  an  inconvenience.  Habit,  however, 
■would  alleviate  it ;  and  some  inconveniences  must  be  cheerfully  submitted  to  by  all  who 
rr«"ivp  ro  act  on  general  principles.  All  other  classes  of  men,  whose  occupation  ex- 
Mtends  to  the  ni^ht  a.-<  well  as  dny,  submit  to  it,  and  find  their  advantage  in  doing  so. 


I'i  *■ 


MEASUREMENT   OF   irME. 


w 


I,  minuter 


circumstance,  such  aa  noon  or  midnight,  but  by  the  motion  of  the  sun 
among  the  Btara.  Time,  so  reckoned,  ia  called  equinoctial  time ;  and  ia 
numerically  the  same,  at  the  same  instant,  m  every  part  of  the  globe. 
Ita  origin  will  be  explained  more  fully  at  a  more  advanced  stage  of  our 
work. 

(149.)  Time  is  an  essential  element  in  astronomical  observation,  in  a 
twofold  point  of  view:  —  Ist-  As  the  representative  of  ang-ular  motion. 
The  earth's  diurnal  motion  bein^-^  uniform,  every  star  deacribes  its  diurnal 
circle  uniformly;  and  the  time  elapsing  between  the  passage  of  the  stars 
in  miccession  across  the  meridian  of  any  observer  becomts,  therei  -re,  a 
direct  measure  of  their  differences  of  right  ascension,  2dly,  Aa  tl  e 
fundamental  element  (f>t  natural  mdependent  variable,  to  use  the  lan- 
guage of  geometers)  in  all  dynamical  theories.  The  great  object  of  as- 
tronomy is  the  determination  of  the  laws  of  the  celestial  motions,  and 
their  reference  to  their  proximate  or  remote  causes.  Now,  the  statement 
of  the  law  of  any  observed  motion  in  a  celestial  object  can  be  no  other 
than  a  proposition  declaring  what  has  been,  is,  and  will  be,  the  real  or 
apparent  situation  of  that  object  at  any  time,  past,  present,  or  future.  To 
compare  such  laws,  therefore,  with  observation,  we  must  possess  a  register 
of  the  observed  situations  of  the  object  in  question,  and  of  the  times  when 
they  were  observed. 

(150.)  The  measurement  of  time  is  performed  by  clocks,  chronometers, 
clepsydras,  and  hour-glasses.  The  two  former  are  alone  used  in  modern 
astronomy.  The  hour-glass  is  a  coarse  and  rude  contrivance  for  measur- 
ing, or  rather  counting  out,  fixed  portions  of  time,  and  is  entirely  disused. 
The  clepsydra,  which  measured  time  by  the  gradual  emptying  of  a  largo 
vessel  of  water  through  a  determinate  orifice,  is  susceptible  of  considera- 
ble exactness,  and  was  the  only  dependence  of  astronomers  before  the 
invention  of  clocks  and  watches.  At  present  it  is  abandoned,  owing  to 
the  greater  convenience  and  exactness  of  the  latter  instruments.  In  one 
case  only  has  the  revival  of  its  use  been  proposed ;  viz.  for  the  accurate 
measuromeut  of  very  small  portions  of  time,  by  the  flowing  out  of  mer- 
cury from  a  small  orifice  in  the  bottom  of  a  vessel,  kept  constantly  full 
to  a  fixed  height.  The  .stream  is  intercepted  at  the  moment  of  noting 
any  event,  and  directed  aside  into  a  receiver,  into  which  it  continues  to 
run,  till  the  moment  of  noting  any  other  event,  when  the  intercepting 
cause  is  suddenly  removed,  the  stream  flows  in  its  original  course,  and 
ceases  to  run  into  the  receiver.  The  weiyht  nf  mercury  received,  com- 
pared with  the  weight  received  in  an  interval  of  time  observed  by  the 
clock,  gives  the  interval  between  the  events  observed.  This  ingenious 
and  simple  method  of  resolving,  with  all  possible  precision,  a  problem 


88 


OUTLINES   OF  ASTRONOMY. 


.  I 


ilf 


of  much  iaiportauoo  in  many  physical  inquiries,  is  due  to  tbo  late  Captain 
Kator.  .-,.  ...i  I"  '  '.   • 

(151.)  The  pendulum  dock,  however,  and  the  balance  watch,  v«'\(!i 
thoso  improvements  and  refinemontH  in  its  structure  which  cnusti-uic  it 
emphatically  a  chronometer,^  arc  the  instrurnonta  on  which  the  astro aomer 
depends  for  his  knowledge  of  the  lapse  of  time.  These  inHtrumcuta  arc 
now  brought  to  such  perfection,  that  an  habitual  irregularity  iu  the  rate 
of  going,  to  the  extent  of  a  single  second  in  twenty-four  Lours  in  two 
consecutive  days,  is  not  tolerated  iu  one  of  good  character;  so  that  any 
interval  of  time  less  than  twenty-four  hours  may  be  certainly  ascortiiinetl 
within  a  few  tenths  of  a  second,  by  their  use.  In  proportion  as  intervals 
are  longer,  the  risk  of  error,  as  well  "as  the  amount  of  error  risked,  be- 
comes greater,  because  the  accidental  errors  of  many  days  may  accumu- 
late ;  and  causes  producing  a  slow  progressive  change  in  the  rate  of  going 
may  subsist  unpcrccived.  It  is  not  safe,  therefore,  to  trust  the  determi- 
nation of  time  to  clocks,  or  watches,  for  many  days  in  succession,  without 
checking  them,  and  ascertaining  their  errors  by  reference  to  natural  events 
which  we  know  to  happen,  day  after  day,  at  equal  intervals.  But  if  this 
be  done,  the  longest  intervals  may  be  fixed  with  the  same  precision  as  the 
shortest ;  since,  in  fact,  it  is  then  only  the  times  intervening  between  the 
Srst  and  the  last  momoiit'  of  such  long  intervals,  and  such  of  those 
periodically  recurring  evcufs  ?;  lopted  for  our  points  of  reckoniug,  as  occur 
within  twenty-four  Iit"*ri  rosrectively  of  either,  that  we  measure  by  arti- 
ficial means.  Tiie  whole  'iayi  are  counted  out  for  us  by  nature ;  the  frac- 
tional parts  only,  at  either  end,  are  measured  by  our  clocks.  To  keep  the 
reckoning  of  the  integer  days  correct,  so  that  none  shall  be  lost  or  counted 
twice,  is  the  object  of  the  calendar.  Chronology  marks  out  the  order  of 
succession  of  events,  and  refers  them  to  their  proper  years  and  days ; 
while  chronometry,  grounding  its  determinations  on  the  precise  observa- 
tion of  such  regularly  periodical  events  as  can  be  conveniently  and  exactly 
subdivided,  enables  us  to  fix  the  rioments  in  vyhich  phenomena  occur,  with 
the  last  degree  of  precision. 

(152.)  In  the  culmination  or  transit  (i.  e.  the  passage  across  the 
meridian  of  an  observer,)  of  every  star  in  the  heavens,  he  is  furnished 
with  .4uch  a  regularly  periodical  natural  event  as  we  allude  to.  Accord- 
ingly, it  is  to  the  transitis  of  the  brightest  and  most  conveniently  situated 
fixed  stars  that  ustronomers  resort  to  ascertain  their  exact  time,  or,  which 
comes  to  the  same  thing,  to  determine  the  exact  amount  of  error  of  their 
clocks. 

(153.)  Before  we  describe  the  instrument  destined  for  the  purpose  of 
'  Xpovof ,  time ;  iitrpeiv,  to  measure. 


PRINCIPLE  OF   TELESCOPIC  SIGHTS. 


89 


observing  such  oulminations,  however,  or  those  iutcndod  for  the  measure- 
ment  of  angular  intervals  in  the  sphere,  it  is  requisite  to  place  clearly 
before  the  reader  the  priuciplo  on  which  the  telisscope  is  applii  1  in  astro- 
nomy to  the  rociflo  dctormiuution  of  a  direction  in  space,  —  thut,  namely, 
of  the  visual  ray  by  which  wo  see  a  star  or  any  other  distant  object. 

(154.)  The  telescope  most  commonly  used  in  astronomy  for  these  pur- 
poses is  the  refracting  telescope,  which  consists  of  an  object-glass  (either 
single,  or  as  ia  now  almost  universal,  double,  forming  what  is  called  in 
optics,  an  acromatic  combination)  A;  a  tube  AB,  int'  '\ich  the  brass 
cell  of  the  object-glass  is  firmly  screwed,  and  au  f^     which  ia 


Fig.  14. 


i 


:^^ 


often  substituted  a  combination  of  glasses  designed  to  increase  the  magni- 
fying power  of  the  telescope,  or  otherwise  give  more  distinctness  of  vision 
cccording  to  optical  principles  which  wo  '':ive  no  ocou.sion  here  to  rofcr  to. 
This  also  is  fitted  into  a  cell,  wliieh  is  screwed  firmly  into  the  end  B  of 
the  tube,  so  that  object-glass,  tube,  and  eye-glass  may  be  considered  as 
forming  one  piece,  invariable  in  the  relative  po.sition  of  its  parts. 

(155)  The  line  PQ  joining  the  centres  of  the  object  and  cyo-glaHses 
and  produced,  is  called  the  axis,  or  line  of  collimation  of  the  I'  Icaeope. 
And  it  is  evident,  that  th?  situation  of  this  line  holds  a  fixod  relation  to 
the  tube  and  its  appctidagcs,  so  I'^nnr  as  the  object  and  eye-glasisos  maintain 
their  fixity  in  this  respect. 

(156.)  Whatever  di.stant  object  E,  this  line  is  directed  to,  an  inverted 
picture  or  imago  of  that  object  F  is  formed  (according  to  the  principles  of 
optics),  in  the  focus  of  the  object-glass,  and  may  there  be  viewed  as  if  it 
icere  a  real  object,  through  the  eye-lens  C,  which  (if  of  short  focus)  ena- 
bles us  to  magnify  it  just  as  such  a  lens  would  magnify  a  material  object 
in  the  same  place. 

(157.)  Now  as  this  imago  is  formed  and  viewed  in  the  air,  being  itself 
immaterial  and  impalpable  —  nothing  prevents  our  placing  in  that  very 
place  F  in  the  axis  of  the  telescope,  a  real,  substantial  object  of  very 
definite  form  and  delicate  make,  such  as  a  fine  metallic  point,  as  of  a 
needle  —  or  better  still,  a  cross  formed  by  two  very  fine  threads  (spider- 
lines),  thin  metallic  wires,  or  lines  drawn  on  glass  intersecting  each  other 
at  right  angles  —  and  whose  intersection  is  all  but  a  mathematical  point. 
If  such  a  point,  wire,  or  cross  be  carefully  placed  and  firmly  fixad  ia  tho 


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90 


OUTLINES  OF  ASTRONOMY. 


exact  focus  F,  both  of  the  object  and  eye-glass,  it  will  be  seen  through  tho 
latter  at  the  same  time,  and  occupying  tJie  same  precise  place  as  the  imago 
of  the  distant  star  E.  The  magnifying  power  of  the  lens  renders  percep* 
tible  the  smallest  deviation  from  perfect  coincidence,  which,  should  it  exist, 
is  a  proof,  that  the  axis  Q  P  is  not  directed  rigorously  towards  Tin.  In  that 
case,  a  fine  motion  (by  means  of  a  screw  duly  applied),  communicated  to 
the  telescope,  will  be  necessary  to  vary  the  direction  of  the  axis  till  the 
coincidence  is  rendered  perfect.  So  precise  is  this  mode  of  pointing  found 
in  practice,  that  the  axis  of  a  telescope  may  be  directed  towards  a  star 
or  other  defirUo  celestial  object  without  an  error  of  more  than  a  few  tenths 
of  a  second  of  angular  measure. 

(158.)  This  application  of  the  telescope  may  be  considered  as  completely 
annihilating  that  part  of  the  error  of  observation  which  might  otherwise 
arise  from  an  erroneous  estimation  of  the  direction  in  which  an  object  lies 
from  the  observer's  eye,  or  from  the  centre  of  the  instrument.  It  is,  in  fact, 
the  grand  source  of  all  the  precision  of  modem  astronomy,  without  which  all 
other  refiuements  in  instrumental  workmanship  would  be  thrown  away ; 
the  errors  capable  of  being  committed  in  pointing  to  an  object,  without 
such  assistance,  being  far  greater  than  what  could  arise  from  any  but  the 
very  coarsest  graduation.'  In  fact,  the  telescope  thus  applied  becomes, 
with  respect  to  angtilar,  what  the  microscope  is  with  respect  to  linear 
dimension.  By  concentrating  attention  on  its  smallest  parts,  and  magni- 
fying into  palpable  intervals  the  minutest  differences,  it  enables  us  not 
only  to  scrutinise  the  form  and  structure  of  the  objects  to  which  it  is 

•  The  honour  of  this  capital  improvement  has  been  successfully  vindicated  by  Der- 
ham  (Phil.  Trans,  xxx.  603)  to  our  young,  talented,  and  unfortunate  countryman  Gas- 
coigne,  from  his  correspondence  with  Crabtree  and  Horrockes,  in  his  (Derham's) 
possession.  The  passages  cited  by  Derham  from  these  letters  leave  no  doubt  that,  so 
early  as  1640,  Gascoigne  had  applied  telescopes  to  his  quadrants  and  sextanis,  with 
threads  in  the  common  focus  of  the  glasses  ;  and  had  even  carried  the  invention  so  far 
as  to  illuminate  the  field  of  view  by  artificial  light,  which  he  found  "  very  helpful  lehen 
the  moon  appeareth  not,  or  it  is  not  otherwise  light  enough."  These  inventions  were 
freely  communicated  by  him  to  Crabtree,  and  through  him  to  his  friend  Horrockes,  the 
(rride  and  boast  of  British  astronomy  ;  both  of  whom  expressed  their  unbounded  admi- 
ration of  this  and  many  other  of  his  delicate  and  admirable  improvements  in  the  art 
of  observation.  Gascoigne,  however,  perished,  at  the  age  of  twenty-three,  at  the 
battle  of  Marstou  Moor ;  and  the  premature  and  sudden  death  of  Horrockes,  at  a  yet 
earlier  age,  will  account  for  the  temporary  oblivion  of  the  invention.  It  was  revived, 
ur  re-invented,  in  1667,  by  Picard  and  Auzout  (Lalande,  Astron.  2310),  after  which  its 
use  became  universal.  Monn,  even  earlier  than  Gascoigne  (in  1635),  had  proposed  to 
substitute  the  telescope  for  plain  sights ;  but  it  is  the  thread  or  wire  utretched  in  the 
focus  with  which  the  image  of  a  star  can  be  brought  to  exact  coincidence,  which 
gives  the  telescope  its  advantage  in  practice  ;  and  the  idea  of  this  does  not  seem  to  have 
occurred  to  Mono.    See  Lalande,  vhri  supri. 


■  t: 


THB  TRANSIT  INSTRUMENT,  i^ 


91 


pointed,  bui  to  refer  their  apparent  places,  with  all  but  geometrical  pre- 
cision, to  the  parts  of  any  scale  with  which  we  propose  to  compare  them. 
(159.)  We  now  return  to  our  subject,  the  determination  of  time  by  the 
transits  or  culminations  of  celestial  objects.  The  instrument  with  which 
such  culminations  are  observed  is  called  a  transit  instrument  It  consists 
of  a  telescope  firmly  fastened  on  a  horizontal  axis  directed  to  the  east  and 
west  points  of  the  horizon,  or  at  right  angles  to  the  plane  of  the  meridian 
of  the  place  of  observation.  The  extremities  of  the  axis  are  formed  into 
cylindrical  pivots  of  exactly  equal  diameters,  which  rest  in  notches  formed 
in  metallic  supports,  bedded  (in  the  case  of  large  instruments)  on  strong 
pieces  of  stone,  and  susceptible  of  nice  adjustment  by  screws,  both  in  a 
vertical  and  horizontal  direction.    By  the  former  adjustment,  the  axis  can 


Fig.  15. 


be  rendered  precisely  horizontal,  by  levelling  it  with  a  level  made  to  rest 
on  the  pivots.  By  the  latter  adjustment  the  axis  is  brought  precisely  into 
the  east  and  west  directions,  the  criterion  of  which  is  furnished  by  the 
observations  themselves  made  with  the  instrument,  in  a  manner  presently 
to  be  explained,  or  by  a  well-defined  object,  called  a  meridian-mark,  origi- 
nally determined  by  such  observations,  and  then,  for  convenience  of  ready 
reference,  permanently  established,  at  a  great  distance,  exactly  in  a  meri' 
dian  line  passing  through  the  central  point  of  the  whole  instrumci>fc.  It 
is  evident,  from  this  description,  that,  if  the  axis,  or  line  of  colliuiation 
of  the  telescope  be  once  well  adjusted  at  right  angles  to  the  axis  of  the 
transit,  it  will  never  quit  the  plane  of  the  meridian,  when  the  instrument 
is  turned  round  on  its  axis  of  rotation. 

(160.)  In  the  focus  of  the  eye-piece,  and  at  right  angles  to  the  length 
of  the  telescope,  is  placed,  not  a  single  cross,  as  in  our  general  explanation 
in  art.  157,  but  a  system  of  one  horizontal  and  several  equidistant  vertical 
threads  or  wires,  (five  or  seven  are  more  usually  employed,)  as  represented 
in  the  annexed  figure,  which  always  appear  in  the  field  of  view,  when 
properly  illuminated,  by  day  by  the  light  of  the  sky,  by  night  by  that  of 
a  lamp  introduced  by  a  contrivance  not  necessary  here  to  explain.     Thu 


OUTLINES  OF  ASTRONOMT. 


"■V<^^   ii'Vj/iJlJ».)>it:.'i.Jj:   j:lf;,. 


Fig.  16. 


■  <!'^x-t.^,:..tr^.,-:.-^<!5*j;... 
'.'.>S.     ■*■■  .-X  ■'?/>V;',,..  ■•  ■■■•.  ■     'r  ;,■.■),  ■ 

' "  •■:  .■■(  vtjri  ■'■ft;  tfv'  j'»:'w  '■:  •■■  :■■'■ 


place  of  this  system  of  wires  may  be  altered  by  adjusting  scitiws,  giving 
it  a  lateral  (horizontal)  motion ;  and  it  is  by  this  means  brought  to  such 
a  position,  that  the  middle  one  of  the  vertical  wires  shall  intersect  t?ie  line 
of  collimation  of  the  telescope,  where  it  is  arrested  and  permanently 
fastened.'  In  this  situation  it  is  evident  that  the  middle  thread  will  bo  a 
visible  representation  of  that  portion  of  the  celestial  meridian  to  which 
the  telescope  is  pointed ;  and  when  )  star  is  seen  to  cross  this  wire  in 
the  telescope,  it  is  in  the  act  of  culminating,  or  passing  the  celestial  meri- 
dian. The  instant  of  this  event  is  noted  by  the  clock  or  chronometerj^ 
which  fo.  IMS  an  indispensable  accompaniment  of  the  transit  instrument. 
For  greater  precision,  the  moment  of  its  crossing  all  the  vertical  threads 
is  noted,  and  a  mean  taken,  which  (since  the  threads  are  equidistant) 
would  give  exactly  the  same  result,  were  all  the  observations  perfect,  and 
will,  of  course,  tend  to  subdivide  and  destroy  their  errors  in  an  average  of 
the  whole  in  the  contrary  case. 

(161.)  For  the  mode  of  executing  the  adjustments,  and  allowing  for 
the  errors  unavoidable  in  the  use  of  this  simple  and  elegnnt  instrument, 
the  reader  must  consult  works  especially  devoted  to  this  department  of 
practical  astronomy.'  We  shall  here  only  mention  one  it  va.t  verifica- 
tion of  its  correctness,  which  consists  in  reversing  the  end  ..  the  axis,  or 
turning  it  east  for  west.  If  this  be  done,  and  it  continue  to  give  the 
same  results,  and  intersect  the  same  point  on  the  meridian  mark,  we  may 
be  sure  that  the  line  of  collimation  of  the  telescope  is  truly  at  right  angles 
to  the  axis,  and  describes  strictly  a  plane,  i.  e.  marks  out  in  the  heavens 
a  great  circle.  In  good  transit  observat'ons,  an  error  of  two  or  three 
tenths  of  a  second  of  time  in  the  moment  of  a  star's  culmination  is  the 
utmost  which  need  be  apprehended,  exclusive  of  the  error  of  the  clock :  in 

*  There  u  no  way  of  bringing  the  true  oplic  axi$  of  the  object-glass  to  coincide 
exactly  with  the  line  of  collimation,  but,  so  long  as  the  object-glass  does  not  shift  or 
ithake  in  its  cell,  any  line  holding  an  invariable  position  with  respect  to  that  axis,  may 
oe  t3Ken  for  the  conventional  or  astronomical  axis  with  equal  effect. 

*  See  Dr.  Pearson's  Treatise  on  Practical  Astronomy.  Also  Bianchi  Sopra  lo 
f<tromentc  do'  Passagi.    Ephem.  di  Milano,  1824. 


MEASUREMENT  OF  AKOULAR   INTERVALS. 


98 


other  words;  a  dock  may  be  compared  with  the  earth's  diurnal  motion  by 
a  single,  observation,  without  risk  of  greater  error.  By  multiplying 
observations,  of  course,  a  yet  greater  degree  of  precision  may  be  obtained. 

(162.)  The  plane  described  by  the  line  of  coUimation  of  a  transit  ought 
to  be  that  of  the  meridian  of  the  place  of  observation.  Tc  ascertain 
whether  it  is  so  or  not,  celestial  observation  must  be  resorted  to.  Now,  as 
the  meridian  is  a  great  circle  passing  through  the  pole,  it  necessarily 
bisects  the  diurnal  circles  described  by  all  the  stars,  all  which  describe  the 
two  semicircles  so  arising  in  equal  intervals  of  12  sidereal  hours  each. 
Hence,  if  we  choose  a  star  whose  whole  diurnal  circle  is  above  the  horizon, 
or  which  never  sets,  and  obsom  the  moments  of  its  upper  and  lower  tran- 
sits across  the  middle  wire  of  the  telescope,  if  we  find  the  two  semidiurnal 
portions  east  and  west  of  the  plane  described  by  the  telescope  to  be 
described  in  precuely  equal  times,  we  may  be  sure  that  plane  is  the  meridian. 

(163.)  The  angular  intervals  measured  by  means  of  the  transit  instru- 
ment and  clock  are  arcs  of  the  equinoctial,  intercepted  between  circles  of 
declination  passing  through  the  objects  observed ;  and  their  measurement, 
in  this  case,  is  performed  by  no  artificial  graduation  of  circles,  but  by  the 
help  of  the  earth's  diurnal  motion,  which  carries  equal  arcs  of  the  equi- 
noctial across  the  meridian,  in  equal  times,  at  the  rate  of  15°  per  sidereal 
hour.    Tn  all  other  cases,  when  we  would  measure  angular  intervals,  it  is 

■.       ,        .^      ...■■^--•-.       .  ^ii'-i.    Fig.  17.   ,■;,.   ,^.j  _.-..      ,,-.i,     .,,;:.:     ,.^; 


a   s 


ianchi  Sopra  lo 


necessary  to  have  recourse  to  circles,  or  portions  of  circles,  constructed  of 
metal  or  other  firm  and  durable  material,  and  mechanically  subdivided  into 
equal  parts,  such  as  degrees,  minutes,  &c.  The  simplest  and  most  obvious 
mode  in  which  the  measurement  of  the  angular  interval  between  two 
directions  in  space  can  be  performed  is  as  follows.  Let  ABCD  be  a 
circle,  divided  into  360  degrees,  (numbered  in  order  from  any  point  0°  in 
the  circumference,  round  to  the  same  point  again,)  and  connected  with  it» 
centre  by  spokes  or  rays,  x,  y,  z^  firmly  united  to  its  circumference  or 


94 


\  i 


OUTLINES  OP  ASTRONOMY. 


limb.  At  the  centre  let  a  circular  hole  be  pierced,  in  which  shall  move 
a  pivot  exactly  fitting  it,  carrying  a  tube,  whose  axis,  ab,  is  exactly 
parallel  to  the  plane  of  the  circle,  or  perpendicular  to  the  pivot;  and  also 
two  arms,  m,  n,  at  right  angles  to  it,  and  forming  one  piece  with  the  tube 
and  the  axis ;  so  that  the  motion  of  the  axis  on  the  centre  shall  carry  the 
tube  and  arms  smoothly  round  the  circle,  to  be  arrested  and  fixed  at  any 
point  we  please,  by  a  contrivance  called  a  clamp.  Suppose,  now,  we 
would  measure  the  antrular  interval  between  two  fixed  objects,  S,  T.  The 
plane  of  the  circle  must  first  be  adjusted  so  as  to  pass  through  them  both, 
and  immoveably  fixed  and  maintained  in  that  position.  This  done,  let  the 
axis  ab  of  the  tube  be  directed  to  one  of  them,  S,  and  damped.  Then 
will  a  mark  on  the  arm  m  point  either  exactly  to  some  one  of  the  divisions 
on  the  limb,  or  between  two  of  them  adjacent.  In  the  former  case,  the 
division  must  be  noted  as  tJie  reading  of  the  arm  m.  In  the  latter,  the 
fractional  part  of  one  whole  interval  between  the  consecutive  divisions  by 
which  the  mark  on  m  turpasses  the  last  inferior  division  must  be  estimated 
or  measured  by  some  mechanical  or  optical  means.  (See  art.  165.)  Thei 
division  and  fractional  part  thus  noted,  and  reduced  into  degrees,  minutes, 
and  seconds,  is  to  be  set  down  as  the  reading  of  the  limb  corresponding 
to  that  position  of  the  tube  a  b,  where  it  points  to  the  object  S.  The 
same  must  then  be  done  for  the  object  T ;  the  tube  pointed  to  it,  and  the 
limb  "  read  off,"  the  position  of  the  circle  remaining  meanwhile  unaltered. 
It  is  manifest,  then,  that,  if  the  lesser  of  these  readings  be  subtracted 
from  the  greater,  their  difference  will  be  the  angular  interval  between  S 
and  T,  as  seen  from  the  centre  of  the  circle,  at  whatever  point  of  the  limb 
the  comn  ;ncement  of  the  graduations  or  the  point  0°  be  situated. 

(164.)  The  very  same  result  will  be  obtained,  if,  instead  of  making  the 


Fig.  18. 


!   ? 


.1 


tube  moveable  upon  the  circle,  we  connect  it  invariably  with  the  latter, 
and  make  both  revolve  together  on  an  axis  concentric  with  the  circle,  and 
Ibrming  one  piece  with  it,  working  in  a  hollow  formed  to  receive  and  fit 
it  in  some  fixed  support.  Such  a  combination  is  represented  in  section  in 
the  annexed  sketch.  T  is  the  tube  or  sight,  fastened,  atpp,  on  the  circle 
AB,  whu^e  axis,  D,  works  in  the  solid  metallic  centring  £,  from  which 


if 


METHODS   OF   READING   OFF. 


95 


origiDates  an  arm,  F,  carrying  at  i  ji  extremity  an  index,  or  other  proper 
mark,  to  point  oat  and  read  off  the  exact  division  of  the  circle  at  B,  the 
point  close  to  it.  It  is  evident  that,  as  the  telescope  and  circle  revolve 
through  any  angle,  the  part  of  the  limb  of  the  latter,  which  by  such 
revolution  is  carried  past  the  index  F,  will  measure  the  angle  described. 
This  is  the  most  usual  mode  of  applying  divided  circles  in  astronomy. 

(165.)  The  index  F  may  either  be  a  simple  pointer,  like  a  clock  hand 
(j^-  «)  J  or  a  vernier  (^fig.  h) ;  or,  lastly,  a  compound  microscope  (Jig. 


Fig.  19. 


Ji.'i 


r^ii^:- 


c 


a  n 


P'  ' 


c),  represented  in  section  in  Jig.  d,  and  furnished  with  a  cross  in  the  com- 
mon focus  of  its  object  and  eye-glass,  moveable  by  a  fine-threaded  screw, 
by  which  the  intersection  of  the  cross  may  be  brought  to  exact  coincidence 
with  the  image  of  the  nearest  of  the  divisions  of  the  circle  formed  in  the 
focus  of  the  object  lens  upon  the  very  same  principle  with  that  explained, 
art.  157  for  the  pointing  of  the  telescope,  only  that  here  the  fiducial  cross 
is  made  moveable ;  and  by  the  turns  and  parts  of  a  turn  of  the  screw 
required  for  this  purpose  the  distance  of  that  division  from  the  original  or 
zero  point  of  the  microscope  may  be  estimated.  This  simple  but  delicate 
contrivance  gives  to  the  reading  off  of  a  circle  a  degree  of  accuracy  only 
limited  by  the  power  of  the  microscope,  and  the  perfection  with  which  a 
screw  can  be  executed,  and  places  the  subdivision  of  angles  on  the  same 
footing  of  optical  certainty  which  is  introduced  into  their  measurement  by 
the  use  of  the  telescope.  >  ;';  -      ;  ;  ■-'  '  ■     '  ■ 

(166.)  The  exactness  of  the  result  thus  obtained  must  depend,  1st,  on 
the  precision  with  which  the  tube  a  b  can  be  pointed  to  the  objects ;  2dly, 
on  the  accuracy  of  graduation  of  the  limb ;  3dly,  on  the  accuracy  with 
which  the  subdivision  of  the  intervals  between  any  two  consecutive  gradu- 
ations can  be  performed.  The  mode  of  accomplishing  the  latter  object 
with  any  required  exactness  has  been  explained  in  the  last  article.  With 
regard  to  the  graduation  of  the  limb,  being  merely  of  a  mechanical  nature. 


96 


OUTLINES  07  ASTRONOMT. 


we  sball  pass  it  without  remark,  further  than  this,  that,  in  the  present 
state  of  instrument-making,  the  amount  of  error  from  this  source  of  inao- 
curacy  is  reduced  within  very  narrow  limits  indeed.'  With  regard  to  the 
first,  it  must  be  obvious  that,  if  the  sights  a  6  be  nothing  more  than 
simple  crosses,  or  pin-holes  at  the  ends  of  a  hollow  tube,  or  an  eye-hole 
at  one  end,  and  a  cross  at  the  other,  no  greater  nicety  in  pointing  can  be 
expected  than  what  simple  vision  with  the  naked  eye  can  command.  But 
if,  in  place  of  these  simple  but  coarse  contrivances,  the  tube  itself  be  con- 
verted into  a  telescope,  having  an  object-glass  at  h,  an  eye-piece  at  a,  and 
a  fiducial  cross  in  their  common  focus,  as  explained  in  art.  1 57 ;  and  if 
the  motion  of  the  tube  on  the  limb  of  the  circle  be  arrested  when  the 
object  is  brought  just  into  coincidence  with  the  intersectional  point  of 
that  cross,  it  is  evident  that  a  greater  degree  of  exactness  may  be  attained 
in  the  pointing  of  the  tube  than  by  the  unassisted  eye,  in  proportion  to 
the  magnifying  power  and  distinctness  of  the  telescope  used. 

(167)  The  simplest  mode  in  which  the  measurement  of  an  angular  in- 
terval can  be  executed,  is  what  we  have  just  described ;  but,  in  strictness, 
this  mode  is  applicable  only  to  terrestrial  angles,  such  as  those  occupied 
on  the  sensible  horizon  by  the  objects  which  surround  our  station, — be- 
cause these  only  remain  stationary  during  the  interval  while  the  telescope 
is  shifted  on  the  limb  from  one  object  to  the  other.  But  the  diurnal 
motion  of  the  heavens,  by  destroying  this  essential  condition,  renders  the 
direct  measurement  of  angular  distance  from  object  to  object  by  this  means 
impossible.  The  same  objection,  however,  does  not  apply  if  we  seek  only 
to  determine  the  interval  between  the  diurnal  circles  described  by  any 
two  celestial  objects.  Suppose  every  star,  in  its  diurnal  revolution,  were 
to  leave  behind  it  a  visible  trace  in  the  heavens, — a  fine  line  of  light,  for 
instance, — then  a  telescope  once  pointed  to  a  star,  so  as  to  have  its  image 
brought  to  coincidence  with  the  intersection  of  the  wires,  would  constantly 
remain  pointed  to  some  portion  or  other  of  this  line,  which  would  there- 
fore continue  to  appear  in  it«  field  as  a  luminous  line,  permanently  inter- 
secting the  same  point,  till  the  star  came  round  again.  From  one  such 
line  to  another  the  telescope  might  be  shifted,  at  leisure,  without  error ; 
and  then  the  angular  interval  between  the  two  diurnal  circles,  in  the  plane 
of  the  telescope's  rotation,  might  be  measured.  Now,  though  we  cannot 
see  the  path  of  a  star  in  the  heavens,  we  can  wait  till  the  star  itself  crosses 
the  field  of  view,  and  seize  the  moment  of  its  passage  to  place  the  inter- 
section of  its  wires  so  that  the  star  shall  traverse  it ;  by  which,  when  the 

'  In  the  great  Ertel  circle  a*.  Pulkova,  the  probable  amount  of  the  accidental  error 
of  division  is  stated  hy  M.  Struve  not  to  exceed  (y-264.  Desc.  de  TObs.  centrale  de 
Pulkova,  p.  147. 


OF  THE  MURAL  CIRCLE. 


4 


telescope  is  well  clamped,  we  equally  well  secure  the  position  of  its  diurnal 
circle  as  if  we  coDtinued  to  see  it  ever  so  long.  The  reading  off  of  the 
limb  may  then  be  performed  at  leisure ;  and  when  another  star  comes 
round  into  the  plane  of  the  circle,  we  may  unclamp  the  telescope,  and  a 
similar  observution  will  enable  us  to  assign  the  place  of  its  diurnal  circle 
on  the  limb :  and  the  observations  may  be  repeated  alternately,  every  day, 
as  the  stars  pass,  till  we  are  satisfied  with  their  result.  ^ 

(168.)  This  is  the  principle  of  the  mural  circle,  which  is  nothing  more 
than  such  a  circle  as  we  have  described  in  att.  163,  firmly  supported,  in 
the  plane  of  the  meridian,  on  a  long  and  powerful  horizontal  axis.  This 
axis  is  let  into  a  massive  pier,  or  wall,  of  stone  (whence  the  name  of  the 
instrument),  and  so  secured  by  screws  as  to  be  capable  of  adjustment 
both  in  a  vertical  and  horizontal  direction ',  so  that,  like  the  axis  of  the 
transit,  it  can  be  maintained  in  the  exact  direction  of  the  east  and  west 
points  of  the  horizon,  the  plane  of  the  circle  being  consequently  truly 
meridional. 

(169.)  The  meridian,  being  at  right  angles  to  all  the  diurnal  circles 
described  by  the  stars,  its  arc  intercepted  between  any  two  of  them  will 
measure  the  least  distance  between  these  circles,  and  will  be  equal  to  the 
difference  of  the  declinations,  as  also  to  the  difference  of  the  meridian 
altitvdes  of  the  objects — at  least  when  corrected  for  refraction.  These 
differences,  then,  are  the  angular  intervals  directly  measured  by  the  mural 
circle.  But  from  these,  supposing  the  law  and  amount  of  refraction 
known,  it  is  easy  to  conclude,  not  their  differences  only,  but  the  quantities 
themselves,  as  we  shall  now  explain. 

(170.)  The  declination  of  a  heavenly  body  is  the  complement  of  its 
distance  from  the  pole.  The  pole,  being  a  point  in  the  meridian,  n  :^:t 
be  directly  observed  on  the  limb  of  the  circle,  if  any  star  stood  exactl' 
therein ;  and  thence  the  polar  distances,  and,  of  course,  the  declinations 
of  all  the  rest,  might  be  at  once  determined.  But  this  not  being  the 
case,  a  bright  star  as  near  the  pole  as  can  be  found  is  selected,  and  observed 
in  its  upper  and  lower  culminations;  that  is,  when  it  passes  the  meridian 
above  and  helow  the  pole.  Now,  as  its  distance  from  the  pole  remains 
the  same,  the  difference  of  reading  off  the  cu'cle  in  the  two  cases  is,  of 
course  (when  corrected  for  refraction),  equal  to  twice  the  polar  distance 
of  the  star;  the  arc  intercepted  on  the  Mmb  of  the  circle  being,  in  this 
case,  equal  to  the  angular  diameter  of  the  star's  diurnal  circle.  In  the 
annexed  diagram,  H  P  0  represents  the  celestial  meridian,  P  the  pole, 
B  R,  A  Q,  CD  the  diurnal  circles  of  stars  which  arrive  on  the  meridian  at 
B,  ^,  and  C  in  their  upper  and  at  K,  Q,  D  in  their  lower  culminations, 
of  which  D  and  Q  happen  above  the  horizon  HO.  P  is  the  pole;  and  if 
7 


98 


OUTLINES  OF  ASTRONOMY. 


Fig.  20.         .^^    ?>'   ri:'.M'":'.v'  -5   --;  A.^  ,■• 


.■.Mil"  vr '<  '  >■ 

■  1  - 

-<■ 

Q 
U 

1 

?''  /?  , 

% 

■'<■  '.'■ 

ir  "l:''  >>*,■.(.  '  ■'> 


■!•  .'7   •■  '."•If-"    - 


/»    ' 


./;;  !;(     „J 


we  suppose  Ajpo  to  be  the  mural  circle,  haying  S  for  its  centre,  bacpd 
will  be  the  points  on  its  circumference  corresponding  to  B  A  C  P  D  iu  the 
heavens.  Now  the  arcs  &a,  bc,bd,  and  cd&re  given  immediately  by  ob- 
servation ;  and  since  C  P  =  P  D,  we  have  also  cp=pdy  and  each  of  them 
=:^cd,  consequently  the  place  of  the  polar  point,  as  it  is  called,  upon 
the  limb  of  the  circle  becomes  known,  and  the  arcs  ph,  pa,  pc,  which 
represent  on  the  circle  the  polar  distances  required,  become  also  known. 

(171.)  The  situation  of  the  pole  star,  which  is  a  very  brilliant  one,  is 
eminently  favourable  for  this  purpose,  being  only  about  a  degree  and  an  half 
from  the  pole ;  it  is,  therefore,  the  star  usually  and  almost  solely  chosen 
for  this  important  purpose ;  the  more  especially  because,  both  its  culmina- 
tions taking  place  at  great  and  not  very  different  altitudes,  the  refractions 
by  which  they  are  affected  are  of  small  amount,  and  differ  but  slightly 
from  each  other,  so  that  their  correction  is  easily  and  safely  applied.  The 
brightness  of  the  pole  star,  too,  allows  it  to  be  easily  observed  in  the  day- 
time. In  consequence  of  these  peculiarities,  this  star  is  one  of  constant 
resort  with  astronomers  for  the  adjustment  and  verification  of  instruments 
of  almost  every  description.  In  the  case  of  the  transit,  for  instance,  it 
furnishes  an  excellent  object  for  the  application  of  the  method  of  testing 
the  meridional  situation  of  the  instrument  described  in  art.  162,  in  fact, 
the  most  advantageous  of  any  for  that  purpose,  owing  to  its  being  the 
most  remote  from  the  zenith,  at  its  upper  culmination,  of  all  bright  stars 
observable  both  above  and  below  the  pole. 

(172.)  The  place  of  the  polar  point  on  the  limb  of  the  mural  circle 
once  determined,  becomes  an  origin,  or  zero  point,  from  which  the  polar 
distances  of  all  objects,  referred  to  other  points  on  the  same  limb,  reckon. 
It  matters  not  whether  the  actual  commencement  0°  of  the  graduations 
stand  there,  or  not ;  since  it  is  only  by  the  differences  of  the  readings 
that  the  arcs  on  the  limb  are  determined ;  and  hence  a  great  advantage  is 


OF  THE   MERIDIAN  CIRCLE. 


<,m 


obtained  in  the  power  of  oommenoing  anew  a  fresh  series  of  observations, 
in  which  a  different  part  of  the  ciroumfereuoe  of  ihe  circle  shall  be 
employed,  and  different  graduations  brought  into  use,  by  which  inequalities 
of  division  may  be  detected  and  neutralized.  This  is  accomplished  prac- 
tically by  detaohing  the  telescope  from  its  old  bearings  on  the  circle,  and 
fixing  it  afresh,  by  screws  or  olampsi  on  a  different  part  of  the  circum- 
ference, -^..l.^i  ,}]»i<-ii-t:.  %  iLit'y  Jt    ■  'I'V  ". ,{  ■    'J;-  fA«:  !,,r' 

(178.)  A  point  on  the  limb  of  the  mural  circle,  not  less  important 
than  the  polar  point,  is  the  horizontal  point,  which,  being  once  known, 
becomes  in  like  manner  an  origin,  or  zero  point,  from  which  altitudes  are 
reckoned.  The  principle  of  its  determination  is  ultimately  nearly  the 
same  with  that  of  the  polar  point.  As  no  star  exists  in  the  celestial 
horizon,  the  observer  must  seek  to  determine  two  points  on  the  limb, 
the  one  of  which  shall  be  precisely  as  far  below  the  horizontal  point  as 
the  other  is  above  it.  For  this  purpose,  a  star  is  observed  at  its  culmina- 
nation  on  one  night,  by  pointing  the  telescope  directly  to  it,  and  the  next, 
by  pointing  to  the  image  of  the  same  star  reflected  in  the  still,  unruffled 
surface  of  a  fluid  at  perfect  rest.  Mercury,  as  the  most  reflective  fluid 
known,  is  generally  chosen  for  that  use.  As  the  surface  of  a  fluid  at  rest 
is  necessarily  horizontal,  and  as  the  angle  of  reflection,  by  ihe  laws  of  optics, 
is  equal  to  that  of  incidence,  this  image  will  be  just  as  much  depressed 
below  the  horizon  as  a  star  itself  is  above  it  (allowing  for  the  difference 
of  refraction  at  the  moment  of  observation).  The  arc  intercepted  on  the 
limb  of  the  circle  between  the  star  and  its  reflected  image  thus  consecu- 
tively observed,  when  corrected  for  refraction,  is  the  double  altitude  of 
the  star,  and  its  point  of  bisection  the  horizontal  point.  The  reflecting 
surface  of  a  fluid  so  used  for  the  determination  of  the  altitudes  of  objects 
b  called  an  artificial  horizon.^ 

(174.)  The  mural  circle  is,  in  fact,  at  the  same  time,  a  transit  instru- 
ment; and,  if  furnished  with  a  propel  system  of  vertical  wires  in  the 
focus  of  its  telescope,  may  be  used  as  such.  As  the  axis,  however,  is 
only  supported  at  one  end,  it  has  not  the  strength  and  permanence  neces< 
sary  for  the  more  delicate  purposes  of  a  transit;  nor  can  it  be  verified,  as 
a  transit  may,  by  the  reversal  of  the  two  ends  of  its  axis,  east  or  west. 


I  By  a  peculiar  and  delicate  manipulation  and  management  of  the  setting  bisection 
and  reading  off  of  the  circle,  aided  by  the  use  of  a  moveable  horizontal  micrometic 
wire  in  the  focus  of  the  object-glass,  it  is  found  practicable  to  observe  a  slow  moving 
star  (as  the  pole  star)  on  one  and  the  tame  night,  both  by  reflection  and  direct  vision, 
sufficiently  near  to  either  culmination  to  give  the  horizontal  point,  withont  tisking  the 
change  of  refraction  in  twenty- four  hours;  so  that  this  source  of  error  is  thus  com- 
pletely eliminated. 


100 


OUTLINES  OF  ASTRONOMT. 


NothiDg,  however,  provents  a  divided  cirole  being  pennftnontly  fkstonod 
on  the  axis  of  a  transit  instrument,  either  near  to  one  of  its  extremities, 
or  close  to  the  tolesoope,  so  as  to  revolve  with  it,  the  reading  c  T  being 
performed  by  one  or  more  miorosoopes  fixed  on  one  of  its  piers.  Sach  an 
instrument  is  called  a  transit  oirole,  or  a  meridian  oirole,  and  serves 
for  the  simultaneous  determination  of  the  right  ascensions  and  polar  dis- 
tances of  objects  observed  with  it ;  the  time  of  transit  being  noted  by  the 
clock,  and  the  circle  being  read  off  by  the  lateral  microscopes.  There  is 
much  advantage,  when  extensive  catalogues  of  small  stars  have  to  be 
formed,  in  this  simultaneous  determination  of  both  their  celestial  co-ordi- 
nates :  to  which  may  be  added  the  facility  of  applying  to  the  meridian 
circle  a  tolesgope  of  any  length  and  optical  power.  The  con>struotion  of 
the  muinl  ^rcle  renders  this  highly  inconvenient,  and  indeed  impracticable 
beyondi  very  moderate  limits. 

(175.)  The  determination  of  the  horizontal  point  on  the  limb  of  an 
instrument  is  of  such  essential  importance  in  astronomy,  that  the  student 
should  be  made  acquainted  with  every  means  employed  for  this  purpose. 
These  are,  the  artificial  horizon,  the  plumb-line,  the  level,  and  the  colli- 
mator. The  artificial  horizon  has  been  already  explained.  The  plurab- 
llne  is  a  fine  thread  or  wire,  to  which  is  suspended  a  weight,  whoso  oscil- 
lations are  impeded  and  quickly  reduced  to  rest  by  plunging  It  in  water. 
The  direction  ultimately  assumed  by  such  a  line,  admitting  its  pe^/ect 
fiexihility,  is  that  of  gravity,  or  perpendicular  to  the  surface  of  still 
water.  Its  application  to  the  purposes  of  astronomy  is,  however,  so  dcU- 
oat«,  and  difficult,  and  liable  to  error,  unless  extraordinary  precautions  are 
taken  in  its  use,  that  it  is  at  present  almost  universally  abandoned,  fot 
the  more  convenient,  and  equally  exact  instrument  the  level. 

(176.)  The  level  is  a  glass  tube  nearly  filled  with  a  liquid,  (spirit  of 
wine,  or  sulphuric  ether,  being  thus  now  generally  used,  on  account  of 


;-3U' ,:  ««iW'1(a>^"'^':!V  :\->  m:-,!*?';h 


Fig.  21. 


■  •»<?/■.( 


smi»^j  Q 


their  extreme  mobility^  and  not  being  liable  to  freeze,)  the  bubble  in 
which,  when  the  tube  is  placed  horizontally,  would  rest  Indifferently  in 
any  part  if  tlie  tube  could  be  mathematically  straight.     But  that  being 


<i< 


DETERMINATION  OF  THE  HORIZONTAL  POINT. 


101 


impossible  to  execute,  and  every  tube  having  some  slight  curvature ;  'f 
the  convex  side  bo  placed  upwards  the  bubble  will  occupy  the  higher  port, 
08  in  the  figure  (where  tbo  curvature  is  purposely  exaggerate  J).  Suppose 
such  a  tube,  as  A  B,  firmly  fastened  on  a  straight  bar,  0  Df  and  marked 
at  a  b,  two  points  distant  by  the  length  of  the  bubble;  then,  if  the 
instrument  bo  so  placed  that  the  bubble  shall  occupy  this  interval,  it  is 
clear  that  C  D  can  have  no  other  than  one  definite  inclination  to  the  hori- 
zon ;  because,  were  it  ever  so  little  moved  one  way  or  other,  the  bubble 
would  shift  its  place,  and  run  towards  the  elevated  side.  Suppose,  now, 
that  we  would  ascertain  whether  any  given  line  P  Q  bo  horizontal ;  let 
the  base  of  the  level  C  D  be  set  upon  it,  and  note  the  points  a  b,  between 
which  the  bubble  is  exactly  contained ;  then  turn  the  level  end  for  end, 
so  that  C  shall  rest  on  Q,  and  D  on  P.  If  then  the  bubble  continue  to 
occupy  the  same  place  between  a  and  5,  it  is  evident  that  F  Q  can  be  no 
otherwise  than  horizontal.  If  not,  the  side  towards  which  the  bubble 
runs  is  highest,  and  must  be  lowered.  Astronomical  levels  ore  furnished 
with  a  divided  scale,  by  which  the  places  of  the  ends  of  the  bubble  can 
be  nicely  marked ;  and  it  is  said  that  they  can  be  executed  with  such 
delicacy,  as  to  indicate  a  single  second  of  angular  deviation  from  exact 
horizontality.  In  such  levels  accident  is  not  trusted  to  to  give  the  requi- 
site curvature.  They  are  ground  and  polished  internally  by  peculiar 
mechanical  processes  of  great  delicacy. 

(177.)  The  mode  in  which  a  level  may  be  applied  to  find  the  horizontal 
point  on  the  limb  of  a  vortical  divided  circle  may  be  thus  explained ;  let 
A  B  be  a  telescope  firmly  fixed  to  such  a  circle,  D  E  F,  and  moveable  in 
one  with  it  on  a  horizontal  axis  C,  which  must  be  like  that  of  a  transit, 
susceptible  of  reversal  (see  art.  161)  and  with  which  the  circle  is  insep- 
arably connected.  Direct  the  telescope  on  some  distant  well-defined  object 
S,  and  bisect  it  by  its  horizontal  wire,  and  in  this  position  clamp  it  fast. 
Let  L  be  a  level  fastened  at  right  angles  to  an  arm,  L  E  F,  furnished  with 
a  microscope,  or  vernier  at  F,  and,  if  we  please,  another  at  E.  Let  this 
arm  be  fitted  by  grinding  on  the  axis  C,  but  capable  of  moving  smoothly 
on  it  without  carrying  it  round,  and  also  of  being  clamped  fast  on  it,  so  as 
to  prevent  it  from  moving  until  required.  While  the  telescope  is  kept 
fixed  on  the  object  S,  let  the  level  be  set  so  as  to  bring  its  bubble  to  the 
marks  a  b,  and  clamp  it  there.  Then  will  the  arm  L  C  F  have  some  cer- 
tain determinate  inclination  (no  matter  what)  to  the  horizon.  In  thib 
position  let  the  circle  be  read  off  at  F,  and  then  let  the  whole  apparatus 
be  reversed  by  turning  its  horizontal  axis  end  for  end,  without  unclamping 
the  level  arm  from  the  axis.  This  done,  by  the  motion  of  the  whole  in- 
strument (level  and  all)  on  its  axis,  restore  the  level  to  its  horizontal  poai- 


102 


OUTLINES   OP  ASTRONOMY. 


tion  with  the  bubble  at  a  h.  Then  we  are  sure  that  tho  telescope  haa 
now  the  same  inclination  to  the  horizon  the  other  wa^,  that  it  had  when 
pointed  to  S,  and  the  reading  off  at  F  will  not  have  been  changed.  Now 
unclamp  the  level,  and  keeping  it  nearly  horizontal,  turn  round  the  circle 
ou  the  axis,  so  as  to  carry  back  the  telescope  through  the  zenith  to  S,  and 
in  that  position  clamp  the  circle  and  telescope  fast.  Then  it  is  evident 
that  an  angle  equal  to  twice  the  zenith  distance  of  S  has  been  moved  over 
by  the  axis  of  the  telescope  from  its  last  position.  Lastly,  without  un- 
clamping  the  telescope  and  circle,  let  the  level  be  once  more  rectified. 
Then  will  the  arm  L  E  F  once  more  assume  the  same  definite  position 
with  respect  to  the  horizon ;  and,  consequently,  if  the  circle  be  again  read 
off,  the  difference  between  this  and  the  previous  reading  must  measure  the 
arc  of  its  circumference  which  has  passed  under  the  point  F,  which  may 
be  considered  as  having  all  the  while  retained  an  invariable  position. 
This  difference,  then,  will  be  the  double  zenith  distance  of  S,  and  its  half 
will  be  the  zenith  distance  simply,  the  complement  of  which  is  its  altitude. 
Thus  the  altitude  corresponding  to  a  given  reading  of  the  limb  becomes 
known,  or,  in  other  words,  the  horizontal  point  on  the  limb  is  ascertained. 
Circuitous  as  this  process  may  appear,  there  is  no  other  mode  of  employ- 
ing the  level  for  this  purpose  which  does  not  in  the  end  come  to  the  same 
thing.  Most  commonly,  however,  the  level  is  used  as  a  mere  fiducial  re- 
ference, to  preserve  a  horizontal  point  once  well  determined  by  other 
means,  which  is  done  by  adjusting  it  so  as  to  stand  level  when  the  tele- 
Fcope  is  truly  horizontal,  and  thus  leaving  it,  depending  on  the  permanence 
of  its  adjustment. 

(178.)  The  last,  but  probably  not  the  least  exact,  as  it  certainly  is,  in 


DETERMINATION   OF  THE  HORIZONTAL   POINT. 


103 


innumerable  cases,  the  most  convenient  means  of  ascertaining  the  horizon- 
tal point,  is  that  afforded  by  the  floating  collimator,  ;n  invention  of  Cap- 
tain Kater,  but  of  vrhich  the  optical  principle  was  first  employed  by  Rit- 
tenhouse,  in  1785,  for  the  purpose  of  fixing  a  definite  direction  in  space 
by  the  emergence  of  parallel  rays  from  a  material  object  placed  in  the 
focus  of  a  fixed  k^os.  This  elegant  instrument  is  nothing  more  than  a 
small  telescope  furnished  with  a  cross-wire  in  its  focus,  and  fastened  hori- 
zontally, or  as  nearly  so  as  may  be,  on  a  flat  iron^a^,  which  is  made  to 
swim  on  mercury,  and  which,  of  course,  will,  when  left  to  itself,  assume 
always  one  and  the  same  invariable  inclination  to  the  horizon.  If  the 
cross-wires  of  the  collimator  be  illuminated  by  a  lamp,  being  in  the  focus 
of  its  object-glass,  the  rays  from  them  will  issue  parallel,  and  will  there- 
Fig.  23. 


fore  be  in  a'  fit  state  to  be  brought  to  a  focus  by  the  object-glass  of  any 
other  telescope,  in  which  they  will  form  an  image  as  if  th^  came  from 
a  celestial  object  in  their  direction,  i.  e.  at  an  altitude  equal  to  their  decli- 
nation. Thus  the  intersection  of  the  cross  of  the  collimator  may  be  ob- 
served as  if  it  were  a  star,  and  that,  however  near  the  two  telescopes  are 
to  each  other.  By  transferring  then,  the  collimator  still  floating  on  a  ves- 
sel of  mercury  from  the  one  side  to  the  other  of  a  circle,  we  are  furnished 
with  two  qtiasi-celestial  objects,  at  precisely  equal  altitudes,  on  opposite 
sides  of  the  centre ;  and  if  these  be  observed  in  succession  with  the  tele- 
scope of  the  circle,*  bringing  its  cross  to  bisect  the  image  of  the  cross  of  the 
collimator  (for  which  end  the  wires  of  the  latter  cross  are  purposely  set 
45°  inclined  to  the  horizon),  the  difference  of  the  readings  on  its  limb 
will  be  twice  the  zenith  distance  of  either ;  whence,  as  in  the  last  article, 
the  horizontal  or  zenith  point  is  immediately  determined.  Another,  and, 
in  many  respects,  preferable  form  of  the  floating  collimator,  in  which  the 
telescope  is  vertical,  and  whereby  the  zenith  point  is  directly  ascertained, 
is  described  in  the  Phil.  Trans.  1828,  p,  257,  by  the  same  author. 

(179.)  By  far  the  neatest  and  most  delicate  application  of  the  princi- 
ple of  collimation  of  Rittenhouae,  however,  is  suggested  by  Benzenberg, 
which  affords  at  once,  and  by  a  single  observation,  an  exact  knowledge 
of  the  nadir  point  of  an  astronomical  circle.     In  this  combination,  the 


104 


OUTLINES  OF  ASTRONOMY. 


::V-«A  ^Jf  j)!j:.;L,i5i'>!>'"j"?a 'Ji.     Fig.  24.        '^  ;-■■-;  ■••fa  ,»:''<;:     .■•''■•v>,?jHT;i| 


!.■  •■    '^    ,  4.    .•  ,  ...    ' 


:,..-;    .,,'j    ■^•.>   .v't^TI     r?>    .  i--*'!v  "  •■ 

!■  J!.;/  ■■••  ';<■.    'v(;f->?'i''(J:-,"    .rt'  ,'  ■• 

.■,■■■   '■,'.■!!     t.-.i    >■;..  .J 
■■,'-      ■•'&    "^C  '■    ■:••.    ■' 

r 


telescope  of  the  circle  is  its  own  collimator.  The  object  observed  is  the 
central  intersectional  cross  of  the  wires  in  its  own  focus  reflected  in  mer- 
cury. A  strong  illumination  being  thrown  upon  the  system  of  wires 
(art.  160)  by  a  lateral  lamp,  the  telescope  of  the  instrument  is  directed 
vertically  downwards  towards  the  surface  of  the  mercury,  as  in  the  figure 
annexed.  The  rays  diverging  from  the  wires  issue  in  parallel  pencils 
from  the  object-glass,  are  incident  on  the  mercury,  and  are  thence  re- 
flected back  (without  losing  their  parallel  character)  to  the  object-glass, 
which  is  therefore  enabled  to  collect  them  again  in  its  focus.  Thus  is 
formed  a  reflected  image  of  the  system  of  cross-wires,  which,  when 
brought  by  the  slow  motion  of  the  telescope  to  exact  boincidence  (inter- 
section upon  intersection)  with  the  real  system  as  seen  in  the  eye-pieoo 
of  the  iustrumei  >.,  indicates  the  precise  and  rigorous  verticality  of  the 
optical  axis  of  the  telescope  when  directed  to  the  nadir  point. 

(180.)  The  transit  and  mural  circle  are  essentially  meridian  instru- 
ments, being  used  only  to  observe  the  stars  at  the  moment  of  their 
meridian  passage.  Independent  of  this  being  the  most  favourable  mo- 
ment for  seeing  them,  it  is  tbnt  in  which  their  diurnal  motion  is  parallel 
to  the  horizon.  It  is  therefore  easier  at  this  time  than  it  could  be  at  any 
other,  to  place  the  telescope  exactly  in  their  true  direction ;  since  their 
apparent  course  in  the  field  of  view  being  parallel  to  the  horizontal  thread 
of  thi'  system  of  wires  therein,  they  may,  by  giving  a  fine  motion  to  the 


I II 


COMPOUND  INSTRUMFNTS  WITH  CO-ORDINATE  CIRCLES.     105 


telescope,  be  brought  to  e..  cu  ooinoidence  with  it,  and  time  may  be 
allowed  to  examine  and  correct  this  coincidence,  if  not  at  first  accurately 
hit,  which  is  the  case  in  no  other  situation.  Generally  speaking,  all 
angular  magnitudes  which  it  is  of  importance  to  ascertain  exactly,  should, 
if  possible,  be  observed  at  their  maxima  or  minima  of  increase  or  dimi- 
nution; because  at  these  points  they  remain  not  perceptibly  changed 
during  a  time  long  enough  to  complete,  and  even,  in  many  cases,  to 
repeat  and  verify,  our  observations  in  a  careful  and  leisurely  manner. 
The  angle  which,  in  the  case  before  us,  is  in  this  predicament,  is  the 
altitude  of  the  star,  which  attains  its  maximum  or  minimum  on  the 
meridian,  and  which  is  measured  on  the  limb  of  the  mural  circle. 

(181.)  The  purposes  of  astronomy,  however,  require  that  an  observer 
should  possess  the  means  of  observing  any  object  not  directly  on  the 
meridian,  but  at  any  point  of  its  diurnal  course,  or  wherever  if  may 
present  itself  in  the  heavens.  Now,  a  point  in  the  sphere  is  determined 
by  reference  to  two  great  circles  at  right  angles  to  each  other ;  or  of  two 
circles,  one  of  which  passes  through  the  pole  of  the  other.  These,  in 
the  language  of  geometry,  are  co-ordinates  by  which  its  situation  is 
ascertained:  for  instance, — on  the  earth,  a  place  is  known  if  we  know 
its  longitude  and  latitude ;  —  in  the  starry  heavens,  if  we  know  its  right 
ascension  and  declination; — in  the  visible  hemisphere,  if  we  know  its 
azimuth  and  altitude,  &c. 

(182.)  To  observe  an  object  at  any  point  of  its  diurnal  course,  we  must 
possess  the  means  of  directing  a  telescope  to  it ;  which,  therefore,  must  be 
capable  of  motion  in  two  planes  at  right  angles  to  each  other ;  and  the 
amount  of  its  angular  motion  in  each  must  be  measured  on  two  circles 
co-ordinate  to  each  other,  whose  planes  must  be  parallel  to  those  in  which 
the  telescope  moves.  The  practical  accomplishment  of  this  conditioi  is 
effected  by  making  the  axis  of  one  of  the  circles  penetrate  that  of  the 
other  at  right  angles.  The  pierced  axis  turns  on  fixed  supports,  while 
the  other  has  no  connection  with  any  external  support,  but  is  s<:stained 
entirely  by  that  which  it  penetrates,  which  is  strengthened  and  enlarged 
at  the  point  of  penetration  to  receive  it.  The  annexed  figure  exhibits 
the  simplest  form  of  such  a  combination,  though  very  far  indeed  from 
the  best  in  point  of  mechanism.  The  two  circles  are  read  off  by  ver- 
niers, or  microscopes;  the  one  attached  to  the  fixed  support  which 
carries  the  principal  axis,  the  other  to  an  arm  projecting  from  that  axis. 
Both  circles  also  are  susceptible  of  being  clamped,  the  clamps  being 
attached  to  the  same  ultimate  bearing  with  which  the  apparatus  for 
reading  off  is  connected. 

(183.)  It  is  manifest  that  such  a  combination,  however  its  principal 


106 


OUTLINES  OF  ASTRONOMT. 


Fig.  26. 


;   );iv;i'i.'r''   >5  >><K^-vv);.; 


-j4fn;i  •! 

■■„.,'. 

i'.  -^V  ' 

.    1..;   'i:i    1 

r. -b.n   i 

, ;  /-• ,  \ ,  ■ 

* 

(.■■■   ,«, 

..      , 

^    .t  .  ■  ) 

••'■'•'     E 

•r>.'v  . 

;  ■  v; 

^ 

'/.U'ij;-   !, 

•>,•>  -.  . 

'.•■■■;;  - 

'■■'■■•) 

■  ■  .   t  A 

■■   -l  h{  ,: 

axis  be  pointed  (provided  that  its  direction  be  invariable,)  will  enable  us 
to  ascertain  the  situation  of  any  object  with  respect  to  the  observer's 
station,  bj  angles  reckoned  upon  two  great  circles  in  the  visible  hemi- 
sphere, one  of  which  has  for  its  poles  the  prolongations  of  the  principal 
axis  or  tho  vanishing  points  of  a  system  of  lines  parallel  to  it,  and  the 
other  passes  always  through  these  poles :  for  the  former  great  circle  is 
the  vanishing  line  of  all  planes  parallel  to  the  circle  A  B,  while  the 
latter,  in  any  position  of  the  instrument,  is  the  vanishing  line  of  all  the 
planes  parallel  to  the  circle  G-  H ;  and  these  two  planes  being,  by  the 
construction  of  the  instrument,  at  right  angles,  the  great  circles,  which 
are  their  vanishing  lines,  must  be  so  too.  Now,  if  two  great  circles  of 
a  sphere  be  at  right  angles  to  each  other,  the  one  will  always  pass 
through  the  other's  poles.  t  i    j- ! 

(184  )  There  are,  however,  but  two  positions  in  which  such  an  appa- 
ratus  can  be  mounted  so  as  to  be  of  any  practical  utility  in  astronomy. 
The  first  is,  when  the  principal  axis  C  D  is  parallel  to  the  earth's  axis, 
and  therefore  points  to  the  poles  of  the  heavens  which  are  the  vanishing 
points  of  all  lines  in  this  system  of  parallels ;  and  when,  of  course,  the 
plane  of  the  circle  A  B  is  parallel  to  the  earth's  equator,  and  therefore 
has  the  equinoctial  for  its  vanishing  circle,  and  measures,  by  its  arcs  r6ad 
off,  hour  angles,  or  differences  of  right  ascension.  In  this  case,  the  great 
circles  in  the  heavens,  corresponding  to  the  various  positions,  which  the 
circle  Q  H  can  be  made  to  assume,  by  the  rotation  of  the  instrument 
round  its  axis  C  D,  are  all  hour-circles;  and  the  arcs  read  off  on  this 
dircle  will  hv  declinations,  or  polar  distances,  or  their  differences. 


THB  EQUATORIAL  INSTRUMENT. 


107 


(185.)  In  thb  posdtion  the  apparatus  assumes  the  uame  of  an  equato- 
rialf  or,  as  it  was  formerly  called,  a  parallactic  instrument.  It  is  a  modi 
convenient  instmment  for  all  such  observations  as  require  an  object  to  be 
kept  long  in  view,  because,  being  once  set  upon  the  object,  it  can  be  fol- 
lowed as  long  as  we  please  by  a  single  motion,  i.  e.  by  merely  turning  the 
whole  apparatus  round  on  its  polar  axis.  For  since,  when  the  telescope 
is  set  on  a  star,  the  angle  between  its  direction  and  that  of  the  polar  axis  is 
equal  to  the  polar  distance  of  the  star,  it  follows,  that  when  turned  about 
its  axis,  without  altering  the  position  of  the  telescope  on  the  circle  Qr  H, 
the  point  to  which  it  is  directed  will  always  lie  in  the  small  circle  of  the 
heavens  coincident  with  the  star's  diurnal  path.  In  many  observations 
this  is  an  inestimable  advantage,  and  one  which  belongs  to  no  other  instru- 
ment. The  equatorial  is  also  used  for  determining  the  place  of  an  un- 
known by  comparison  with  that  of  a  known  object,  in  a  manner  to  be 
described  in  the  fifth  chapter.  °  The  adjustments  of  the  equatorial  are 
somewhat  complicated  and  difficult.  They  are  best  performed  in  this 
manner: — 1st,  Follow  the  pole  star  round  its  whole  diurnal  course,  by 
which  it  will  become  evident  whether  the  polar  axis  is  directed  above  or 
below,  to  the  right  or  to  the  left,  of  the  true  pole, — and  correct  it  accord- 
ingly (without  any  attempt,  during  this  process,  to  correct  the  errors,  if 
any,  in  the  position  of  the  declination  axis).  2dly,  after  the  polar  axis  is 
thus  brought  into  adjustment,  place  the  plane  of  the  declination  circle  in 
or  near  the  meridian ;  and,  having  there  secured  it,  observe  the  transits 
of  several  known  stars  of  widely  different  declinations.  If  the  intervals 
between  these  transits  correspond  to  the  known  differences  of  right  ascen- 
sions of  the  stars,  we  may  be  sure  that  the  telescope  describes  a  true 
meridian,  and  that,  therefore,  the  declination  axis  is  truly  perpendicular 
to  the  polar  one ;  —  if  not,  the  deviation  of  the  intervals  from  this  law 
will  indicate  the  direction  and  amount  of  the  deviation  of  the  axis  in 
question,  and  enable  us  to  correct  it.'     '  ' 

(186.)  A  very  great  improvement  has,  within  a  few  years  of  the  present 
time,  been  introduced  into  the  construction  of  the  equatorial  instrument. 
It  consists  in  applying  a  clockwork  movement  to  turn  the  whole  instru- 
ment round  upon  its  polar  axis,  and  so  to  follow  the  diurnal  motion  of 
any  celestial  object,  without  the  necessity  of  the  observer's  manual  inter- 
vention.    The  driving  power  is  the  descent  of  u  ^;feight  which  communi- 

*  See  Littrow  on  the  Adjustment  of  the  Equatorial  (Mem.  Ast.  Soc.  vol.  ii.  p.  45), 
where  formulae  are  given  for  ascertaining  the  amount  and  direction  of  all  the  misad- 
justments  simultaneously.    But  the  practical  observer,  who  wishes  to  avoid  bewilder 
ing  himself  by  doing  two  things  at  once,  had  better  proceed  as  recommended  in  th« 
text. 


108 


OUTLINES   OP  ASTRONOMY. 


\m 


oates  motion  to  a  train  of  wheelwork,  and  thus,  uUimatelj,  to  the  poUr 
axis,  while,  at  the  same  time,  its  too  sm/i  descent  is  controlled  and  regu- 
lated to  the  exact  and  uniform  rate  required  to  give  that  axis  one  turn  in 
24  hours,  by  connecting  it  with  a  regulating  clock,  or  (which  is  found 
preferable  in  practice)  by  exhausting  all  the  superfluous  energy  of  the 
driving  power,  by  causing  it  to  overcome  a  regulated  friction.  Artists 
have  thus  succeeded  in  obtaining  a  perfectly  smooth,  uniform,  and  regula- 
ble motion,  which,  when  so  applied,  serves  to  retain  any  object  on  which  the 
telescope  may  be  set,  commodiously,  in  the  centre  of  the  field  of  view  for 
whole  hours  in  succession,  leaving  the  attention  of  the  observer  undis- 
tracted  by  having  a  mechanical  movement  to  direct,  and  with  both  his 
hands  at  liberty.  ..  ,  ;  J  ..l-lAis  '<.:ia  in:  .  ,.-,'n-;vf..;  '^i'',;iiM!'i'r;i    ).-.  ^ ;  •■al 

(187.)  The  other  position  in  which  such  a  compound  apparatus  as  we 
have  described  in  art.  182  may  be  advantageously  mounted,  is  that  in 
which  the  principal  axis  occupies  a  vertical  position,  and  the  one  circle,  A 
B,  consequently  corresponds  to  the  celestial  horizon,  and  the  other,  G  H, 
to  a  vertical  circle  of  the  heavens.  The  angles  measured  on  the  former 
are  therefore  azimuOis,  or  differences  of  azimuth,  and  those  of  the  latter 
zenith  distances,  or  altitudes,  according  as  the  graduation  commences  from 
the  upper  point  of  its  limb,  or  from  one  90°  distant  from  it.  It  is  there- 
fore known  by  the  name  of  an  azimuth  and  altitude  instrument.  The 
vertical  position  of  its  principal  axis  is  secured  either  by  a  plumb-line 
suspended  from  the  upper  end,  which,  however  it  be  turned  round,  should 
continue  always  to  intersect  one  and  the  same  fiducial  mark  near  its  lower 
extremity,  or  by  a  level  fixed  directly  across  it,  whose  bubble  ought  not 
to  shift  its  place,  on  moving  the  instrument  in  azimuth.  The  north  or 
south  point  on  the  horizontal  circle  is  ascertained  by  bringing  the  vertical 
circle  to  coincide  with  the  plane  of  the  meridian,  by  the  same  criterion  by 
which  the  azimuthal  adjustment  of  the  transit  is  performed  (art.  162), 
and  noting,  in  this  position,  the  reading  off  of  the  lower  circle ',  or  by  the 
following  process. 

(188.)  Let  a  bright  star  be  observed  at  a  considerable  distance  to  the 
east  of  the  meridian,  by  bringing  it  on  the  cross  wires  of  the  telescope. 
In  this  position  let  the  horizontal  circle  be  read  off,  and  the  telescope 
securely  clamped  on  the  vertical  one.  When  the  star  has  passed  the 
meridian,  and  is  in  the  descending  point  of  its  daily  course,  let  it  be  fol- 
lowed by  moving  the  whole  instrument  round  to  the  west,  without,  how- 
ever, unclamping  the  telescope,  until  it  comes  into  the  field  of  view ;  and 
until,  by  continuing  the  horizontal  motion,  the  star  and  the  cross  of  the 
wires  come  once  more  to  coincide.  In  this  position  it  is  evident  the  star 
must  have  the  same  precise  altitude  above  the  western  horizon,  that  it  had 


ALTITUDE  AND  AZIMUTH  INSTRUMENT. 


109 


at  the  moment  of  the  first  obseiration  above  the  eastern.  At  this  point 
let  the  motion  be  arrested,  and  the  horizontal  circle  be  again  read  off. 
The  difference  of  the  readings  will  be  the  azimuthal  arc  described  in  the 
interval.  Now,  it  is  evident  that  when  the  altitudes  of  any  star  are  equal 
on  either  side  of  the  meridian,  its  azimuths,  whether  reckoned  both  from 
the  north  or  both  from  the  south  point  of  the  horizon,  must  also  be  equal, 
—  consequently  the  north  or  south  point  of  the  horizon  must  bisect  the 
azimuthal  arc  thus  determined,  and  will  therefore  become  known.      '     -' 

(189.)  This  method  of  determining  the  north  and  south  points  of  a 
horizontal  circle  is  called  the  "method  of  equal  altitudes,"  and  is  of  great 
and  constant  use  in  practical  astronomy.  If  we  note,  at  the  moments  of 
the  two  observations,  the  time,  by  a  clock  or  chronometer,  the  instant 
half-way  between  them  will  be  the  moment  of  the  star's  meridian  passage, 
which  may  thus  be  determined  without  a  transit;  and,  vice  versd,  the 
error  of  a  clock  or  chronometer  may  by  this  process  be  discovered.  For 
this  last  purpose,  it  is  not  necessary  that  our  instrument  should  be  pro- 
vided with  a  horizontal  circle  at  all.  Any  means  by  which  altitudefB  can 
be  measured  will  enable  us  to  determine  the  moments  when  the  same  star 
arrives  at  equal  altitudes  in  the  eastern  and  western  halves  of  its  diurnal 
course ;  and,  these  once  known,  the  instant  of  meridian  passage  and  the 
error  of  the  clock  become  also  known. 

(190.)  Thus  also  a  meridian  line  may  be  drawn  and  a  meridian  mark 
erected.  For  the  readings  on  the  north  and  south  points  on  the  limb  of 
the  horizontal  circle  being  known,  the  vertical  circle  may  be  brought  ex- 
actly into  the  plane  of  the  meridian,  by  setting  it  to  that  precise  reading. 
This  done,  let  the  telescope  be  depressed  to  the  north  horizon,  and  let  the 
point  intersected  there  by  its  cross-wires  be  noted,  and  a  mark  erected 
there,  and  let  the  same  be  done  for  the  south  horizon.  The  line  joining 
these  pointi<  is  a  meridian  line,  passing  through  the  centre  of  the  hori- 
zontal circle.     The  marks  may  be  made  secure  and  permanent  if  required. 

(191.)  One  of  the  chief  purposes  to  which  the  altitude  and  azimuth 
circle  is  applicable  is  the  investigation  of  the  amount  and  law*^  of  refrac- 
tion. For,  by  following  with  it  a  circumpolar  star  which  passes  the 
zenith,  and  another  which  grazes  the  horizon,  through  their  whole  diurnal 
course,  the  exact  apparent  form  of  their  diurnal  orbits,  or  the  ovals  into 
which  their  circles  are  distorted  by  refraction,  can  be  traced ;  and  their 
deviation  from  circles,  being  at  every  moment  given  by  the  nature  of  the 
observation  in  the  direction  in  which  the  refraction  itself  takes  place  (i.  e. 
in  altitude),  is  made  a  matter  of  direct  observation. 

(192.)  The  zenith  sector  and  the  theodolite  are  peculiar  modifications' 
of  the  altitude  and  azimuth  instruqient.    The  former  is  adapted  for  the 


110 


/iiM't   OUTLINES  OF  ASTRONOMY. 


very  exact  observation  of  stars  in  or  near  the  zenith,  bj  giving  a  great 
length  to  the  vertical  azus,  and  suppresfling  all  the  circumference  of  the 
vertical  circle,  except  a  few  degrees  of  its  lower  part,  by  which  a  great 
length  of  radius,  and  a  consequent  proportional  enlargement  of  the  divi- 
sions of  its  arc,  is  obtained.  The  latter  is  especially  devoted  to  the  mea- 
sures of  horizontal  angles  between  terrestrial  objects,  in  which  the  telescope 
never  requires  to  be  elevated  more  than  a  few  degrees,  and  in  which, 
therefore,  the  vertical  circle  is  either  dispensed  with,  or  executed  on  a 
smaller  scale,  and  with  less  delicacy;  while,  on  the  other  hand,  great  care 
b  bestowed  on  securing  the  exact  perpendicularity  of  the  plane  of  the 
telescope's  motion,  by  resting  its  horizontal  axis  on  two  supports  like  the 
piers  of  a  transit-instrument,  which  themselves  are  firmly  bedded  on  the 
spokes  of  the  horizontal  circle,  and  turn  with  it.      w;!  /«/,■   .  i^;  ^<     ] 

(198.)  The  next  instrument  we  shall  describe  is  one  by  whose  aid  the 
angular  distance  of  any  two  objects  may  be  measured,  or  the  altitude  of 
a  single  one  determined,  either  by  measuring  its  distance  from  the  visible 
horizon  (such  as  the  sea-offing,  allowing  for  its  dip),  or  from  its  own  reflec- 
tion on  thd  surface  of  mercury.  It  is  the  sextant,  or  quadrant,  commonly 
jailed  Hadley'a,  from  its  reputed  inventor,  though  the  priority  of  inven- 
tion belongs  undoubtedly  to  Newton,  whose  claims  to  the  gratitude  of  the 
navigator  are  thus  doubled,  by  his  having  furnished  at  once  the  only 
theory  by  which  his  vessel  can  be  securely  guided,  and  the  only  instru- 
ment which  has  ever  been  found  to  avail,  in  applying  that  theory  to  its 
nautical  uses.'      irr  - 

(194.)  The  principle  of  this  instrument  is  the  optical  property  of  re- 
flected rays,  thus  announced: — "The  angle  between  the  first  and  last 
directions  of  a  ray  which  has  suffered  two  reflections  in  one  plane  is  equal  to 
twice  the  inclination  of  the  reflecting  surfaces  to  each  other.  Let  A  B  be 
the  limb  or  graduated  arc,  of  a  portion  of  a  circle  60°  in  extent,  but 
divided  into  120  equal  parts.  On  the  radius  C  B  let  a  silvered  plane  glass 
D  be  fixed,  at  right  angles  to  the  plane  of  the  circle,  and  on  the  moveable 
radius  C  E  let  another  such  silvered  glass,  C,  be  fixed.  The  glass  D  is 
permanently  fixed  parallel  to  A  C,  and  only  one  half  of  it  is  silvered,  the 
other  half  allowing  objects  to  be  seen  through  U.  The  glass  C  is  wholly 
silvered,  and  its  plane  is  parallel  to  the  length  of  the  moveable  radius  C  E, 

*  Newton  communicated  it  to  Dr.  Halley,  who  suppressed  it.  The  description  of 
the  instrumttnt  was  found,  after  the  death  of  Halley,  among  his  papers,  in  Newton's 
own  handwriting,  by  his  executor,  who  communicated  the  papers  to  the  Royal  Society, 
twenty-five  years  after  Newton's  death,  and  eleven  after  the  publication  of  Hadley's 
invention,  which  might  be,  and  probably  was,  independent  of  any  knowledge  of  New- 
ton'b,  though  Hutton  insinuates  the  contrary. 


■A'? 


THE  SEXTANT. 


Ill 


at  the  extremity  E  of  which  a  vernier  is  phused  to  read  off  the  divisions 
of  the  limb.  On  the  radius  A  C  is  set  a  telescope  F,  through  which  any 
object,  Q,  may  be  seen  by  direct  rays  which  pass  through  the  unsilvered 
portion  of  the  glass  D,  while  another  object,  P,  is  seen  through  the  same 
telescope,  by  rays,  which,  after  reflection  at  0,  have  been  thrown  upon  the 
silvered  part  of  D,  and  are  thence  direc'ied  by  a  second  refieetion  into  the 


xil 


Fig.  26. 


!     /; 


I  i' 


telescope.  The  two  images  so  formed  will  both  be  seen  in  the  field  of 
view  at  once,  and  by  moving  the  radius  C  E  will  (if  the  reflectors  be  truly 
perpendicular  to  the  plane  o"  the  circle)  meet  and  pass  over,  without  oblit- 
erating each  other.  The  motion,  however,  is  arrested  when  they  meet, 
and  at  this  point  the  angle  included  between  the  direction  C  P  of  one  ob- 
ject, and  F  Q  of  the  other,  is  twice  the  angle  E  C  A  included  between  the 
fixed  and  moveable  radii  0  A,  C  E.  Now,  the  graduations  of  the  limb 
beitfg  purposely  made  only  half  as  distant  as  would  correspond  to  degrees, 
the  arc,  A  E,  when  read  off,  as  if  the  graduations  were  whole  degrees,  will, 
in  fact,  read  double  its  real  amount,  and  therefore  the  numbers  so  read  off 
will  express,  not  the  angle  EGA,  but  its  double,  the  angle  subtended  by 
the  objects.  .  ■ 

(195.)  To  determine  the  exact  distances  between  the  stars  by  direct 
observation  is  comparatively  of  little  service ;  but  in  nautical  astronomy 
the  measurement  of  their  distances  from  the  moon,  and  of  their  altitudes, 
is  of  essential  importance ;  and  as  the  sextant  requires  no  fixed  support^ 
but  can  be  held  in  the  hand,  and  used  on  ship-board,  the  utility  of  the 
instrument  becomes  at  once  obvious.  For  altitudes  at  sea,  as  no  level, 
plumb-line,  or  artificial  horizon  can  be  used,  the  sea-offing  affords  the  only 
resource ;  and  the  image  of  the  star  observed,  seen  by  reflection,  is  brought 
to  coincide  with  the  boundary  of  the  sea  seen  by  direct  rays.  Thus  the 
altitude  above  the  sea-line  is  found ;  and  this  corrected  for  the  dip  of  the 
horizon  (art.  23)  gives  the  true  altitude  of  the  star.     On  land,  an  artifi- 


112 


OUTLINES   OF  A8TR0N0MT. 


oial  horizon  may  be  used  (art.  173),  and  the  consideration  of  dip  is  ren- 
dered unnecessary. 

(106.)  The  adjustments  of  the  sextant  are  simple.  They  consist  in 
fixing  the  two  reflectors,  the  one  on  the  revolving  radius  C  E,  the  other 
on  the  fixed  ono  GB,  so  as  to  have  their  planes  perpendicular  to  the 
plane  of  the  circle,  and  parallel  to  each  other,  when  the  reading  of  the 
instrument  i.s  zero.  This  adjustment  in  the  latter  respect  is  of  little 
moment,  aa  its  effect  is  to  produce  a  constant  error,  whose  amount  is 
readily  ascertained  by  bringing  the  two  images  of  one  and  the  same  star 
or  other  distant  object  to  coincidence;  when  the  instrument  ought  to  read 
zero,  and  if  it  does  not,  the  angle  which  it  does  read  is  the  zero  correction 
and  must  be  subtracted  from  all  angles  measured  with  the  sextant.  The 
former  adjustments  are  essential  to  be  maintained,  and  are  performed  by 
small  screws,  by  whose  aid  either  or  both  the  glasses  may  be  tilted  a  little 
one  way  or  another  until  the  direct  and  reflected  images  of  a  vertical  line  (a 
plumb-line)  can  be  brought  to  coincidence  over  their  whole  extent,  so  as 
to  form  a  single  unbroken  straight  line,  whatever  be  the  position  of  the 
moveable  arm,  in  the  middle  of  the  field  of  view  of  the  telescope,  whose 
Axis  is  carefully  adjusted  by  the  optician  to  parallelism  with  the  plane  of 
the  limb.  In  practice  it  is  usual  to  leave  only  the  reflector  D  on  the  fixed 
radius  adjustable,  that  on  the  moveable  being  set  to  great  nicety  by  the 
maker.  In  this  case  the  best  way  of  making  the  adjustment  is  to  view  a 
pair  of  lines  crossing  each  other  at  right  angles  (ono  being  horizontal,  the 
other  vertical)  through  the  telescope  of  the  instrument,  holding  the  plane 
of  its  limb  vertical,  —  then  having  brought  the  horizontal  line  and  its 
reflected  imago  to  coincidence  by  the  motion  of  the  radius,  the  two 
images  of  the  vertical  arm  must  be  brought  to  coincidence  by  tilting  one 
way  or  other  the  fixed  reflector  D  by  means  of  an  adjusting  screw,  with 
which  every  sextant  is  provided  for  that  purpose.  When  both  lines  coin- 
cide in  the  centre  of  the  field,  the  adjustment  is  correct.  I 

(197.)  The  reflecting  circle  is  an  instrument  destined  for  the  same  uses 
as  the  sextant,  but  more  complete,  the  circle  being  entire,  and  the  divi- 
sions carried  all  round.  It  is  usually  furnished  with  three  verniers,  so  as 
to  admit  of  three  distinct  readings  ofl',  by  the  average  of  which  the  error 
of  graduation  and  of  reading  is  reduced.  This  is  altogether  a  very  refined 
and  elegant  instrument. 

(198.)  We  must  not  conclude  this  part  of  our  subject  without  mention 
of  the  "  principle  of  repetition ;"  an  invention  of  Borda,  by  which  the 
error  of  graduation  may  be  diminished  to  any  degree,  and,  practically 
speaking,  annihilated.  Let  P  Q  be  two  objects  which  we  may  suppose 
fixed,  for  purposes  of  mere  explanation,  and  let  K  L  be  a  telescope  move- 


PRINCIPLE   or  REPETITION. 


lis 


lers,  so  as 


able  on  0,  the  coididoq  axis  of  two  circles,  A  M  L  and  a  be,  of  which  the 
former,  A  M  L,  is  absolutely  fixed  in  the  plane  of  the  objects,  and  carries 
the  graduations,  and  the  latter  is  freely  moveable  on  the  axis.  The  tele- 
scope is  attached  permanently  to  the  latter  circle,  and  moves  with  it.  An 
arm  0  o  A  carries  the  index,  or  vernier,  which  reads  off  the  graduated 
limb  of  the  fixed  circle.  This  arm  is  provided  with  two  clamps,  by  which 
it  can  be  temporarily  connected  with  either  circle,  and  detached  at 
pleasure.  Suppose,  now,  the  telescope  directed  to  P.  Clamp  the  index 
arm  0  A  to  the  inner  circle,  and  unclamp  it  from  the  outer,  and  read  off. 
Then  carry  the  telescope  round  to  the  other  object  Q.  In  so  doing,  the 
inner  circle,  and  the  index-arm  which  is  clamped  to  it,  will  also  be  carried 
round,  over  an  arc  A  B,  on  the  graduated  limb  of  the  outer,  equal  to  the 
angle  P  0  Q.  Now  clamp  the  index  \^  the  outer  circle,  and  unclamp  the 
inner,  and  read  off :  the  difference  of  readings  will  of  course  measure  the 
angle  POQ;  but  the  result  will  be  liable  to  two  sources  of  error — that 
of  graduation  and  that  of  observation,  both  which  it  is  our  object  to  get 
rid  of.  To  this  end  transfer  the  telescope  back  to  P,  without  unclamping 
the  arm  from  the  outer  circle;  then,  having  made  the  bisection  of  P, 
clamp  the  arm  to  b,  and  unclamp  it  from  6,  and  again  transfer  the  tele- 
scope to  Q,  by  which  the  arm  will  now  be  carried  with  it  to  C,  over  a 
second  arc,  B C,  equal  to  the  angle  POQ.  Now  again  read  off;  then 
will  the  difference  between  this  reading  and  the  original  one  measure 
tici'cc  the  angle  POQ,  affected  with  both  errors  of  observation,  but  only 
with  the  same  error  of  graduation  as  be/ore.  Let  this  process  be  re- 
peated as  often  as  we  please  (suppose  ten  times) ;  then  will  the  final  arc 
A  B  C  D  read  off  on  the  circle  be  ten  times  the  required  angle,  affected 
by  the  joint  errors  of  all  the  ten  observations,  but  only  by  the  same  con- 
stant error  of  graduation,  which  depends  on  the  initial  and  final  readings 
off  alone.  Now  the  errors  of  observation,  when  numerous,  tend  to 
8 


(  1 


114 


» 1 


0UILINI8  OF  ASTRONOMT. 


balaaoe  aod  destroy  one  toother ;  so  that,  if  suffioiently  multiplied,  their 
influenoo  will  disappear  from  the  result.  There  remains,  then,  ouly  the 
constant  error  of  graduation,  which  comes  to  be  divided  in  the  final  result 
by  the  number  of  observations,  and  is  therefore  diminished  in  ila  influence 
to  one  tenth  of  its  possible  amount,  or  to  less  if  need  bo.  The  abstract 
beauty  and  advantage  of  this  principle  seem  to  be  counterbalanced  in 
practice  by  some  unknown  cause,  which,  probably,  must  be  sought  for  in 
imperfect  clamping. 

(199.)  Micrometers  are  instruments  (as  the  name  imports')  for  measur- 
ing, with  great  precision,  small  angles,  not  exceeding  a  few  minutes,  or  at 
most  a  whole  degree.  They  are  very  various  in  construction  and  principle, 
nearly  all,  however,  depending  on  the  exceeding  delicacy  with  which  space 
can  be  subdivided  by  the  turns  and  parts  of  a  turn  of  fine  screws.  Thus 
— in  thejparallel  wire  micrometer,  two  parallel  threads  (spider's  lines  arc 
j;enerally  used)  stretched  on  sliding  frames,  one  or  both  moveable  by 


Fig.  28. 


screws  in  a  direction  perpendicular  to  tint  of  the  toreads,  are  placed  in 
the  common  focus  of  the  object  and  cyo-gIas«es  of  a  telescope,  and  brought 
by  the  motion  of  the  screws  exactly  to  cover  the  two  extremities  of  the 
image  of  any  small  object  seen  in  the  telescope,  as  the  diameter  of  a 
planet,  &o.,  the  angular  distance  between  which  it  is  required  to  measure. 
This  done,  the  threads  are  closed  up  by  turning  one  of  the  screws  till  tbey 
exactly  '  ver  each  other,  and  the  number  of  turns  and  parts  of  a  turn 
required  gives  the  interval  of  the  threads,  which  must  be  coilverted  into 
angular  measure,  either  by  actual  calculation  from  the  linear  m^risure  of 
the  threads  of  the  screw  and  the  focal  length  of  the  object  glus^,  or 
experiuientally,  by  measuring  the  image  of  a  known  object  ;  v.: '^  % 
known  distance  (as  a  foot-rule  at  a  hundred  yards,  &c.)  and  theieiore  buI> 
tending  a  known  angle. 

(200.)  The  duplication  of  the  image  of  an  object  by  optical  means 
furnishes  a  va)«i;iblo  and  fertile  resource  in  micrometry.  Suppose  by  any 
optical  contrivf^tic*  the  single  image  A  of  any  object  can  be  converted  into 
two,  exactly  eq<ial  cr '.  s'milar,  .IB,  at  a  distance  from  one  another, 

,    i..  '  Mi«fp«f,  sma)' J /itrptiK,  to  measure.  . 


M^-'i    V  I  ■  't 


MICROMETFl^S. 
Fig.  29. 


115 


OX) 


dependent  (by  some  mechanical  movement)  on  the  will  of  'a  obser\  )r, 
and  in  anj  ruqu^'fed  dlrootion  from  one  another.  As  these  oai  therefoio, 
bo  nado  to  n|,()ri'aoh  to  or  recede  from  each  other  at  pleasure,  ley  may 
be  b"  u^\l  iu  tli  j,  first  place  to  approach  till  they  touch  one  auotber  ou 
oTiu  ti\uQ,  a.4  at  A  'J,  and  then  being  made  by  continuing  the  motion  to 
oross  uuu  ouch  on  the  opposite  side,  as  A  D,  it  is  evident  that  the 
ntity  of  movement  required  to  produce  the  change  from  one  conuict 
to  the  other,  i/um/ormf  will  mecuure  the  double  diameter  of  the  object  A. 
(201.)  Innumerable  optical  combinations  may  be  devised  to  operate 
such  duplication.  The  chief  and  most  important  (from  its  recent  appli- 
cations,) is  the  heliometer,  in  which  the  image  is  divided  by  bisecting  the 
(hject-fflaas  of  the  telescope,  and  making  its  two  halves,  set  in  sepante 
brass  frames,  slide  laterally  on  each  other,  as  A  B,  the  motion  being 


Fig.  80. 


'^.-'iduced  and  measured  by  a  screw.  Each  half,  by  the  laws  of  optics, 
forms  its  own  image  (somewhat  blurred,  it  is  true,  by  diffraction,')  in  its 
own  axis;  and  thus  two  equal  and  similar  images  are  formed  side  by  side 
in  the  focus  of  the  eye-piece,  which  may  be  made  to  approach  and  recede 
by  the  motion  of  the  «urev.  and  thus  afford  the  means  of  measurement 
as  above  described. 
(202.)  Double  refraction  through  crystallized  media  affords  another 

'  This  miglit  be  curc<d,  though  at  nn  eapenw  of  light,  by  limiting  each  half  to  a 
circuiar  space  by  diaphragms,  as  repr«aMU«d  by  the  dotted  lines. 


116 


OUTLINES  OP  ASTRONOMY. 


means  of  accomplishing  tho  same  end.  Without  going  into  the  intrica- 
cies of  this  difficult  branch  of  optics,  it  will  suffice  to  state  that  objects 
viewed  through  certain  crystals  (as  Iceland  spar,  or  quartz)  appear 
double,  two  images  equally  distinct  being  formed,  whose  angular  distance 
from  each  other  varies  from  nothing  (or  perfect  coincidence,)  up  to  a 
certain  limit,  according  to  tJie  direction  with  respect  to  a  certain  fixed 
line  in  tJie  crystal,  called  its  optical  axis.  Suppose,  then,  to  take  the 
simplest  case,  that  the  eye-lens  of  a  telescope,  instead  of  glass,  were 
formed  of  such  a  crystal  (say  of  quartz,  which  may  be  worked  as  well  or 
better  than  glass,)  and  of  a  splierical  form,  so  as  to  offer  no  difference 
when  turned  about  on  its  centre,  other  than  the  inclination  of  its  optical 
axis  to  the  visual  ray.  Then  when  that  axis  coincides  with  the  line  of 
collimation  of  the  object-glass,  one  image  only  will  be  seen,  but  when 
made  to  revolve  on  an  axis  perpendicular  to  that  line,  two  will  arise, 
opening  gradually  out  from  each  other,  and  thus  originating  the  desired 
duplication.  In  this  contrivance,  the  angular  amount  of  the  rotation  of 
the  sphere  affords  the  necessary  datum  for  determining  the  separation  of 
the  images. 

(203.)  Of   all  methods  which  have  been   proposed,   however,   the 
simplest  and  most  unobjectionable  would  appear  to  be  the  following.    It 

Fig.  81. 


is  well  known  to  every  optical  student,  that  two  prisms  of  glass,  a  flint 
and  a  crown,  may  be  opposed  to  each  other,  so  as  to  produce  a  colourless 
deflection  of  parallel  rays.  An  object  seen  through  such  a  compound  or 
achromatic  prism,  will  be  seen  simply  deviated  in  direction,  but  in  no 
way  otherwise  altered  or  distorted.  Let  such  a  prism  be  constructed 
with  its  surfaces  so  nearly  parallel  that  the  total  deviation  produced  iu 
traversing  them  shall  not  exceed  a  small  amount  (say  5'.)  Let  this  bo 
cut  in  half,  and  from  each  half  let  a  circular  disc  be  formed,  and  cemented 


MICROMETERS. 


117 


on  a  circular  plate  of  parallel  glass,  or  otherwise  sustained,  close  to  and 
concentric  with  the  other  by  a  framework  of  metal  so  light  as  to  inter- 
cept but  a  small  portion  of  the  light  which  passes  on  the  outside  (as  in 
the  annexed  figure,)  where  the  dotted  lines  represent  the  radii  sustaining 
one,  and  the  undotted  those  carrying  the  other  disc.  The  whole  must 
be  so  mounted  as  to  allow  one  disc  to  revolve  in  its  own  plane  behind 
the  other,  fixed,  and  to  allow  the  amount  of  rotation  to  be  read  off.  It 
is  evident,  then,  that  when  the  deviations  produced  by  the  two  discs 
conspire,  a  total  deviation  of  10'  will  be  effected  on  all  the  light  which 
has  passed  through  them ;  that  when  they  oppose  each  other,  the  rays 
will  emerge  undeviated,  and  that  in  intermediate  positions  a  deviation 
varying  from  0  to  10',  and  calculable  from  the  angular  rotation  of  the 
one  disc  on  the  other,  will  arise.  Now,  let  this  combination  be  applied 
at  such  a  point  of  the  cone  of  rays,  between  the  object-glass  and  its 
focus,  that  the  discs  shall  occupy  exactly  half  the  area  of  its  section. 
Then  will  half  the  light  of  the  object  lens  pass  undeviated  —  the  other 
half  deviated,  as  above  described;  and  thus  a  duplication  of  image, 
variable  and  measureable  (as  required  for  micrometric  measurement)  will 
occur.  If  the  object-glass  be  not  very  large,  the  most  convenient  point 
of  its  application  will  be  externally  before  it,  in  which  case  the  diameter 
of  the  discs  will  be  to  that  of  the  object-glass  as  707  :  1000  j  or  (allow- 
ing for  the  spokes)  about  as  7  to  10. 

(204.)  The  Position  Micrometer  is  simply  a  straight  thread  or  wire 
w)''.ch  is  carried  round  by  a  smooth  revolving  motion,  in  the  common  focus 
of  the  object  and  eye-glasses,  in  a  plane  perpendicular  to  the  axis  of  the 
telescope.  It  serves  to  determine  the  situation  with  respect  to  some  fixed 
line  in  the  field  of  view,  of  the  line  joining  any  two  objects  or  points  of 
an  object  seen  in  that  field  —  as  two  stars,  for  instance,  near  enough  to  be 
seen  at  once.  For  this  purpose  the  moveable  thread  is  placed  so  as  to 
cover  both  of  them,  or  stand,  as  may  best  be  judged,  parallel  to  their  line 
of  junction.  And  its  angle,  with  the  fixed  one,  is  then  read  off  upon  a 
small  divided  circle  exterior  to  the  instrument.  When  such  a  micrometer 
is  applied  (as  it  most  commonly  is)  to  an  equatorially  mounted  telescope, 
the  zero  of  its  position  corresponds  to  a  direction  of  the  wire,  such  as,  pro- 
longed, will  represent  a  circle  of  declination  in  the  heavens  —  and  the 
"angles  of  position"  so  read  off  are  reckoned  invariably  from  one  point, 
and  in  one  direction,  viz.,  north,  following,  south,  preceding ;  so  that  0° 
position  corresponds  to  the  situation  of  an  object  exactly  north  of  that  as- 
sumed as  a  centre  of  reference, —  90°  to  a  situation  exactly  eastward  or 
following;  180°  exactly  so?f/A;  and  270°  e^SiQil^  west,  ot  preceding  va. 
the  order  of  diurnal  movement. 


r  fi 


118 


OUTLINES  OF  ASTfiONOMT. 


CHAPTER  IV. 
OP    GEOGRAPHY. 

OF  THE  FIGURE  OF  THE  EARTH. — ITS  EXACT  DIMENSIONS. — ^ITS  FORM — 
THAT  OF  EQUILIBRIUM  MODIFIED  BY  CENTRIFUGAL  FORCE.  —  VARIA- 
TION OF  GRAVITY  ON  ITS  SURFACE.  —  STATICAL  AND  DYNAMICAL 
MEASURES  OF  GRAVITY. — THE  PENDULUM.  —  GRAVITY  TO  A  SPHE- 
ROID.— OTHER  EFFECTS  OF  THE  EARTH's  ROTATION. — TRADE  WINDS. 

—  DETERMINATION   OF  GEOGRAPHICAL  POSITIONS.  —  OF  LATITUDES. 

—  OF  LONGITUDES.  —  CONDUCT  OF  A  TRIGONOMETRICAL  SURVEY. — 
OF  MAPS.  —  PROJECTIONS  OF  THE  SPHERE.  —  MEASUREMENT  OF 
HEIGHTS  BY  THE  BAROMETER. 


(205.)  Geography  is  not  only  the  most  important  of  the  practical 
branches  of  knowledge  to  which  astronomy  is  applied,  but  it  is  also, 
theoretically  speaking,  an  essential  part  of  the  latter  science.  The  earth 
being  the  general  station  from  which  we  view  the  heavens,  a  knowledge 
of  the  local  situation  of  particular  stations  on  its  surface  is  of  great  con- 
sequence, when  we  come  to  inquire  the  distances  of  the  nearer  heavenly 
bodies  from  us,  as  concluded  from  observations  of  their  parallax  as  well 
as  on  all  other  occasions,  where  a  difference  of  locality  can  be  supposed  to 
influence  astronomical  results.  We  propose,  therefore,  in  this  chapter,  to 
explain  the  principles,  by  which  astronomical  observation  is  applied  to 
geographical  determinations,  and  to  give  at  the  same  time  an  outline  of 
geography  so  far  as  it  is  to  be  consideired  a  part  of  astronomy. 

(206.)  Geography,  as  the  word  imports,  is  a  delineation  or  description 
of  the  earth.  In  its  widest  sense,  this  comprehends  not  only  the  delinea- 
tion of  the  form  of  its  continents  and  seas,  its  rivers  and  mountains,  but 
their  physical  condition,  climates,  and  products,  and  their  appropriation 
by  communities  of  men.  With  physical  and  political  geography,  how- 
ever, we  have  no  concern  here.  Astronomical  geography  has  for  its 
objects  the  exact  knowledge  of  the  form  and  dimensions  of  the  earth,  the 
parts  of  its  surface  occupied  by  sea  and  land,  and  the  configuration  of  the 
surface  of  the  latter,  regarded  as  protuberant  above  the  ocean,  and  broken 


THB  FIGURE   OF   THE   EARTH. 


119 


into  the  various  forms  of  mountain,  table  land,  and  valley ;  neither  should 
the  form  of  the  bed  of  the  ocean,  regarded  as  a  continuation  of  the  sur< 
face  of  the  land  beneath  the  water,  be  left  out  of  consideration :  we 
know,  it  is  true,  very  little  of  it ;  but  this  is  an  ignorance  rather  to  be 
lamented,  and,  if  possible,  remedied,  than  acquiesced  in,  inasmuch  as 
there  are  many  very  important  branches  of  inquiry  which  would  be 
greatly  advanced  by  a  better  acquaintance  with  it.  ,    . 

(207.)  With  regard  to  the  figure  of  the  earth  as  a  whole,  we  have 
already  shown  that,  speaking  loosely,  it  may  be  regarded  as  spherical ;  but 
the  reader  who  has  duly  appreciated  the  remarks  in  art.  22  will  not  be  at 
a  loss  to  perceive  that  this  result,  concluded  from  observations  not  suscep- 
tible of  much  exactness,  and  embracing  very  small  portions  of  the  surface 
at  once,  can  only  be  regarded  as  a  first  approximation,  and  may  require  to 
be  materially  modified  by  entering  into  minutiae  before  neglected,  or  by 
increasing  the  delicacy  of  our  observations,  or  by  including  in  their  extent, 
larger  areas  of  its  surface.  For  instance,  if  it  should  turn  out  (as  it  will), 
on  minuter  inquiry,  that  the  true  figure  is  somewhat  elliptical,  or  flattened, 
in  the  manner  of  an  orange,  having  the  diameter  which  coincides  with  the 
axis  about  3^i;th  part  shorter  than  the  diameter  of  its  equatorial  circle; 
— this  is  so  trifling  a  deviation  from  the  spherical  form  that,  if  a  model 
of  such  proportions  were  turned  in  wood,  and  laid  before  us  on  a  table, 
the  nicest  eye  or  hand  would  not  detect  the  flattening,  since  the  difference 
of  diameters,  in  a  globe  of  fifteen  inches,  would  amount  only  to  ^^^th  of 
an  inch.  In  all  common  parlance,  and  for  all  ordinary  purposes,  then,  it 
would  still  be  called  a  globe ;  while,  nevertheless,  by  careful  measurement, 
the  difference  would  not  fail  to  be  noticed;  and,  speaking  strictly,  it  would 
be  termed,  not  a  globe,  but  an  oblate  ellipsoid,  or  spheroid,  which  is  the 
name  appropriated  by  geometers  to  the  form  above  described. 

(208.)  The  sections  of  such  a  figure  by  a  plane  are  not  circles,  but 
ellipses ;  so  that,  on  such  a  shaped  earth,  the  horizon  of  a  spectator  would 
nowhere  (except  at  the  poles)  be  exactly  circular,  but  somewhat  elliptical. 
It  is  easy  to  demonstrate,  however,  that  its  deviation  from  the  circulur 
form,  arising  from  so  slight  an  "  ellipticity  "  as  above  supposed,  would  be 
quite  imperceptible,  not  only  to  our  eye-sight,  but  to  the  test  of  the  dip- 
sector  ;  so  that  by  that  mode  of  observation  we  should  never  be  led  to 
notice  so  small  a  deviation  from  perfect  sphericity.  How  we  are  led  to 
this  conclusion,  as  a  practical  result,  will  appear,  when  we  have  explained 
the  means  of  determining  with  accuracy  the  dimensions  of  the  whole,  or 
any  part  of  the  earth. 

(209.)  As  we  cannot  grasp  the  earth,  nor  recede  from  it  far  enough  to 
view  it  at  once  as  a  whole,  and  compare  it  with  a  known  standard  of  mca- 


120 


OUTLINES   OF  ASTRONOMY. 


sure  in  any  degree  commensurate  to  its  own  size,  but  can  only  creep  about 
upon  it,  and  apply  our  diminutive  measures  to  comparatively  small  parts 
of  its  vast  surface  in  succession,  it  becomes  necessary  to  supply,  by  geo- 
metrical reasoning,  the  defect  of  our  physical  powers,  and  from  a  delicate 
and  careful  measurement  of  such  small  parts  to  conclude  the  form  and 
dimensions  of  the  whole  mass.  This  would  present  little  difficulty,  if  we 
were  sure  the  earth  was  strictly  a  sphere,  for  the  proportion  of  the  cir- 
cumference of  a  circle  to  its  diameter  being  known  (viz,  that  of  3-1415926 
to  1  0000000),  we  have  only  to  ascertain  the  length  of  the  entire  circum- 
ference of  any  great  circle,  such  as  a  meridian,  in  miles,  feet,  or  any  other 
standard  units,  to  know  the  diameter  in  units  of  the  same  kind.  Now, 
the  circumference  of  the  whole  circle  is  known  as  soon  as  we  know  the 
exact  length  of  any  aliquot  part  of  it,  such  as  1°  or  ^g^th  part ;  and  this, 
being  not  more  than  about  seventy  miles  in  length,  is  not  beyond  the 
limits  of  very  exact  measurement,  and  could,  in  fact,  be  measured  (if  we 
knew  its  exact  termination  at  each  extremity)  within  a  very  few  feet,  or, 
indeed,  inches,  by  methods  presently  to  be  particularized. 

(210.)  Supposing,  then,  we  were  to  begin  measuring  with  all  due  nicety 
from  any  station,  in  the  exact  direction  of  a  meridian  and  go  measuring 
on,  till  by  some  indication  we  were  informed  that  we  had  accomplished  an 
exact  degree  from  the  point  we  set  out  from,  our  problem  would  then  be 
at  once  resolved.  It  only  remains,  therefore,  to  inquire  by  what  indica- 
tions we  can  be  sure,  1st,  that  we  liave  advanced  an  exact  degree  ;  and, 
2dly,  that  we  have  been  measuring  in  the  exact  direction  of  a  great  circle. 

(211.)  Now  the  earth  has  no  landmarks  on  it  to  indicate  degrees,  nor 
traces  inscribed  on  its  surface  to  guide  us  in  such  a  course.  The  compass, 
though  it  affords  a  tolerable  guide  to  the  mariner  or  the  traveller,  is  far 
too  uncertain  in  its  indications,  and  too  little  known  in  its  laws,  to  be  of 
any  ase  in  such  an  operation.  We  must,  therefore,  look  outwards,  and 
refer  our  situation  on  the  surface  of  our  globe  to  natural  marks,  external 
to  it,  and  which  are  of  equal  permanence  and  stability  with  the  earth 
itself.  Such  marks  are  afforded  by  the  stars.  By  observations  of  their 
meridian  altitudes,  performed  at  any  station,  and  from  their  known  polar 
distances,  we  conclude  the  height  of  the  pole ;  and  since  the  altitude  of 
the  pole  is  equal  to  the  latitude  of  the  place  (art.  119)  the  same  obser- 
vations give  the  latitudes  of  any  stations  where  we  may  establish  the 
requisite  instruments.  When  our  latitude  then,  is  found  to  have  dimin- 
i:jhed  a  degree,  we  know  that,  provided  we  have  kept  to  the  meridian,  we 
have  described  one  three  hundred  and  sixtieth  part  of  the  earth's  circum- 
ference. 

(212.)  The  direction  of  the  meridian  may  be  secured  at  every  instant 


FIGURE   OF  THE   EARTH. 


121 


by  the  observations  described  in  art.  162,  188 ;  and  although  local  diffi- 
culties may  oblige  us  to  deviate  in  our  measurement  from  this  exact  direc- 
tion, yet  if  we  keep  a  strict  account  of  the  amount  of  this  deviation,  a 
very  simple  calculation  will  enable  us  to  reduce  our  observed  measure  to 
its  meridional  value. 

(213.)  Such  is  the  principle  of  that  most  important  geographical  ope- 
ration, the  measurement  of  an  arc  of  the  meridian.  In  its  detail,  however, 
a  somewhat  modified  course  must  be  followed.  An  observatory  cannot  be 
mounted  and  dismounted  at  every  step ;  so  that  we  cannot  identify  and 
measure  an  exact  degree  neither  more  nor  less.  But  this  is  of  no  conse- 
qii3nce,  provided  we  know  with  equal  precision  Tiow  much,  more  or  less, 
we  have  measured.  In  place,  then,  of  measuring  this  precise  aliquot 
part,  we  take  the  more  convenient  method  of  measuring  from  one  good 
observing  station  to  another,  about  a  degree,  or  two  or  three  degrees,  as 
the  case  may  be,  or  indeed  any  determinate  angular  interval  apart,  and 
determining  by  astronomical  observation  the  precise  diflFerence  of  latitudes 
between  the  stations. 

(214.)  Again,  it  is  of  great  consequence  to  avoid  in  this  operation 
every  source  of  uncertainty,  because  an  error  committed  in  the  length  of 
a  single  degree  will  be  multiplied  360  times  in  the  circumference,  and 
nearly  115  times  in  the  diameter  of  the  earth  concluded  from  it.  Any 
error  which  may  affect  the  astronomical  determination  of  a  star's  altitude 
will  be  especially  influential.  Now,  there  is  still  too  much  uncertainty 
and  fluctuation  in  the  amount  of  refraction  at  moderate  altitudes,  not  to 
make  it  especially  desirable  to  avoid  this  source  of  error.  To  effect  this, 
we  take  care  to  select  for  observation,  at  the  extreme  stations,  some  star 
which  passes  through  or  near  the  zeniths  of  both.  The  amount  of  refrac- 
tion, within  a  few  degrees  of  the  zenith,  is  very  small,  and  its  fluctuations 
and  uncertainty,  in  point  of  quantity,  so  excessively  minute  as  to  be 
utterly  inappreciable.  Now,  it  is  the  same  thing  whether  we  observe  the 
pole  to  bo  raised  or  depressed  a  degree,  or  the  zenith  distance  of  a  star 
when  on  a  meridian  to  have  changed  by  the  same  quantity  (fig.  art.  128). 
If  at  one  station  we  observe  any  star  to  pass  through  the  zenith,  and  at 
the  other  to  pass  one  degree  south  or  north  of  the  zenith,  we  are  sure  that 
the  geographical  latitudes,  or  the  altitudes  of  the  pole  at  the  two  stations, 
must  differ  by  the  same  amount. 

(215.)  Granting  that  the  terminal  points  of  one  degree  can  be 
ascertained,  its  length  may  be  measured  by  the  methods  which  will  be 
presently  described,  as  we  have  before  remarked,  to  within  a  very  few  feet. 
Now,  the  error  which  may  be  committed  in  fixing  each  of  these  terminal 
points  cannot  exceed  that  which  may  be  committed  in  the  observation  of 


122 


OUTLINES   OF  ASTRONOMY. 


the  zenith  distance  of  a  star  properly  situated  for  the  purpose  in  question. 
This  error,  with  proper  care,  can  hardly  exceed  half  a  second.  Supposing 
we  grant  the  possibility  of  ten  feet  of  error  in  the  length  of  each  degree 
in  a  measured  arc  of  five  degrees,  and  of  half  a  second  in  each  of  the 
zenith  distances  of  one  star,  observed  at  the  northern  and  southern  sta- 
tions, and,  lastly,  suppose  all  these  errors  to  conspire,  so  as  to  tend  all  of 
them  to  give  a  result  greater,  or  all  less,  than  the  truth,  it  will  appear, 
by  a  very  easy  proportion,  that  the  whole  amount  of  error  which  would 
be  thus  entailed  on  an  estimate  of  the  earth's  diameter,  as  concluded 
from  such  a  measure,  would  not  exceed  1147  yards,  or  about  two  thirds 
of  a  mile,  and  t^his  is  ample  allowance. 

(216.)  This,  however,  supposes  that  the  form  of  the  earth  is  that  of  a 
perfect  sphere,  and,  in  con&equenoe,.  the  lengths  of  its  degrees  in  all  parts 
precisely  equal.  But,  when  we  come  to  compare  the  measures  of  meri- 
dional arcs  made  in  various  parts  of  the  globe,  the  results  obtained, 
although  they  agree  sufficiently  to  show  that  the  supposition  of  a  spherical 
figure  is  not  very  remote  from  the  truth,  yet  exhibit  discordances  far 
greater  than  what  we  have  shown  to  be  attributable  to  error  of  observation, 
and  which  render  it  evident  that  the  hypothesis,  in  strictness  of  its  word- 
ing, is  untenable.  The  following  table  exhibits  the  lengths  of  arcs  of  the 
meridian  (astronomically  determined  as  above  described),  expressed  in 
British  standard  feet,  as  resulting  from  actual  measurement  made  with  all 
possible  care  and  precision,  by  commissioners  of  various  nations,  men  of 
the  first  eminence,  supplied  by  their  respective  governments  with  the  best 
instruments,  and  furnished  with  every  facility  which  could  tend  to  ensure 
a  successful  result  of  their  important  labours.  The  lengths  of  the  degrees 
in  the  last  column  are  derived  from  the  numbers  set  down  in  the  two 
preceding  ones  by  simple  proportion,  a  method  not  quite  exact  when  the 
arcs  are  large,  but  sufficiently  so  for  our  purpose. 


% 


I 


FiaURE  OF  THE  EARTH. 


128 


Country. 

Latitude  of 
Middle  of  Ars. 

Are 
measured. 

Measured 

Length  In 

Feet. 

Moan 
liength  of 
the  Dejjree 
at  the  Mid- 
dle Lati- 
tude In 
Feet. 

Sweden,*  A  B   • 

+  66" 

20' 

lO"-0 

l" 

37' 

19"-6 

593277 

365744 

Sweden,  A 

+  66 

19 

37 

0 

67 

30-4 

351832 

367086 

Russia,  A 

+  58 

17 

37 

3 

36 

5-2 

1309742 

365368 

Russia,  B 

+  56 

3 

555 

8 

2 

289 

2937439 

365291 

Prussia,  B 

+  64 

58 

260 

1 

30 

200 

551073 

366420 

Denmark,  B 

+  54 

8 

13-7 

1 

31 

53-3 

559121 

365087 

Hanover,  A  B 

+  52 

32 

16-6 

2 

0 

57-4 

736426 

365300 

England,  A 

+  52 

35 

45 

3 

57 

131 

1442953 

364971 

England,  B 

+  52 

2 

19-4 

2 

50 

23-6 

1036409 

364951 

France,  A 

+  46 

52 

2 

8 

20 

0.3 

3040606 

364872 

France,  A  B 

+44 

51 

2-5 

12 

22 

12-7 

4509832 

364572 

Rome,  A 

+42 

59 

— 

2 

9 

47 

787919 

364262 

America,  A 

+  39 

12 

^^^ 

1 

28 

45-0 

638100 

363786 

India,  A  B 

+  16 

8 

21-5 

15 

57 

40-7 

5794598 

363044 

India,  A  B 

+  12 

32 

20-8 

1 

34 

56-4 

574318 

362956 

Peru,  A  B 

—  1 

31 

0-4 

3 

7 

3-5 

1131050 

363626 

Cape  of  Good  Hope,  A 

—33 

18 

30 

1 

13 

17-5 

445506 

364713 

Cape  of  Good  Hope,  B 

—35 

43 

200 

3 

34 

34-7 

1301993 

364060 

It  is  evident  from  a  mere  inspection  of  the  second  and  fifth  columns  of 
this  table,  that  tJie  measured  length  of  a  degree  increases  with  the  lati- 
tude, being  greatest  near  the  poles,  and  least  near  the  equator.  Let  ns 
now  consider  what  interpretation  is  to  be  put  upon  this  conclusion,  as 
regards  the  form  of  the  earth. 

(217.)  Suppose  we  held  in  our  hands  t*  model  of  the  earth  smoothly 
turned  in  wood,  it  would  be,  as  already  observed,  so  nearly  spherical,  that 
neither  by  the  eye  nor  the  touch,  unassisted  by  instruments,  could  v  e 
detect  any  deviation  from  that  form.  Suppose,  too,  we  were  debarred 
from  measuring  directly  across  from  surface  to  surface  in  different  direc 
tions  with  any  instrument,  by  which  we  might  at  once  ascertain  whether 


•  The  astronomers  by  whom  these 
lows  :  — 

Sweden,  A  B-^  Svanberg. 
Sweden,  A — Maupertuis. 
Russia,  A — Struve. 
Russia,  B  —  Struve,  Tenner, 
Prussia  —  Bessel,  Bayer. 
Denmaric  — Schumacher. 
Hanover  —  Gauss. 
England  —  Roy,  Kater. 
France,  A —  Lacaille,  Cassini. 


measurements  were  executed  were  as  fol 


France,  A  B  -^  De'ambre,  Mechain. 
Rome  —  Bospovich. 
America  —  Mason  and  Dixon. 
India,  1st  — Lambton. 
India,  2d  —  Lambton,  Everest, 
Peru  —  Lacondamine,  Bouguer. 
Cape  of  Good  Hope,  A — Lacaille. 
Cape  of  Good  Hope,  B —  Maclear. 
— Aitr.  Nachr.  574 


124 


OUTLINES   OP  ASTRONOMY. 


one  diameter  were  longer  than  another ;  how,  then,  we  may  ask,  are  we 
to  ascertain  whether  it  is  a  true  sphere  or  not  ?  It  is  clear  that  we  have 
no  resource,  but  to  endeavour  to  discover,  by  some  nicer  means  than 
simple  inspection  or  feeling,  whether  the  convexity  of  its  surface  is  the 
same  in  every  part;  and  if  not,  where  it  is  greatest,  and  where  least. 
Suppose,  then,  a  thin  plate  of  metal  to  be  cut  into  a  concavity  at  its  edge, 
so  as  exactly  to  fit  the  surface  at  A :  let  this  now  be  removed  from  A, 
and  applied  successively  to  several  other  parts  of  the  surface,  taking  care 
to  keep  its  plane  always  on  a  great  circle  of  the  globe,  as  here  represented. 
If,  then,  we  find  any  position,  B,  in  which  the  light  can  enter  in  the 
middle  between  the  globe  and  platCj  or  any  other,  C,  where  the  latter  tilts 
by  pressure,  or  admits  the  light  vnder  its  edges,  we  are  sure  that  the  c%ir- 
vature  of  the  surface  at  B  is  less,  and  at  C  greater,  than  at  A. 

(218.)  What  we  here  do  by  the  application  of  a  metal  plate  of  deter- 
minate length  and  curvature,  we  do  on  the  earth  by  the  measurement  of 
a  degree  of  variation  in  the  altitude  of  the  pole.  Curvature  of  a  surface 
is  nothing  but  the  continual  deflection  of  its  tangent  from  one  fixed  direc- 
tion as  we  advance  along  it.  When,  in  the  same  measured  distance  of 
advance  we  find  the  tangent  (rhich  answers  to  our  horizon)  to  have 
shifted  its  position  with  respect  to  c  fixed  direction  in  space,  (such  as  the 
axis  of  the  heavens,  or  the  line  joiniLg  the  earth's  centre  and  some  given 
star,)  more  in  one  part  of  the  earth's  nieridian  than  in  another,  we  con- 
clude, of  necessity,  that  the  curvature  of  the  surface  at  the  former  spot  is 
greater  than  at  the  latter ;  and  vice  versd,  when,  in  order  to  produce  the 
same  change  of  horizon  with  respect  to  the  pole  (suppose  1°)  we  require 
to  travel  over  a  longer  measured  space  at  one  point  than  at  another,  we 
assign  to  that  point  a  less  curvature.  Hence  we  conclude  that  the  curva- 
ture of  a  meridional  section  of  the  earth  is  sensihly  greater  at  the  equa^ 
lor  than  towards  the  poles ;  or,  in  other  words,  that  the  earth  is  not 
spherical,  but  flattened  at  the  poles,  or,  which  comes  to  the  same,  protu- 
berant at  the  equator. 


FiaURE  OF  THE  EARTH. 


125 


(210.)  Let  N  ABDEF  represent  a  meridional  section  of  the  earth,  C 
its  centre,  and  N  A,  B  D,  G  E,  arcs  of  a  meridian,  each  corresponding  to 
one  degree  of  difference  of  latitude,  or  to  one  degree  of  variation  in  the 
meridian  altitude  of  a  star,  as  referred  to  the  horizon  of  a  spectator  travel- 
ling along  the  meridian.  Let  wN,  a  A,  6B,  dJ),  gG,  cE,  be  the  respec- 
tive directions  of  the  plumb-line  at  the  stations  N,  A,  B,  D,  G,  E,  of 
which  we  will  suppose  N  to  be  at  the  pole  and  E  at  the  equator;  then 
will  the  tangents  to  the  surface  at  these  points  respectively  be  perpen- 
dicular to  these  directions;  and,  consequently,  if  each  pair,  viz.  roN  and 
a  A,  5B  and  dl),  gG  and  cE,  be  prolonged  till  they  intersect  each  other 
(at  the  points  a;,  y,  z),  the  angles  No;  A,  ByD,  GzE,  will  each  be  one 
degree,  and,  therefore,  all  equal ;  so  that  the  small  curvilinear  arcs  N  A, 
B  D,  Gr  E,  may  be  regarded  as  arcs  of  circles  of  one  degree  each,  descnbed 
about  X,  1/,  z,  as  centres.  These  are  what  in  geometrj  are  called  centres 
of  curvature,  and  the  radii  xN  or  xA,  i/B  or  i/ J),  zQ  or  zE,  represent 

Fig.  83. 


I 


i 


radii  of  curvature,  by  which  the  curvatures  at  those  points  arc  deter- 
mined and  measured.  Now,  as  the  arcs  of  different  circles,  which  sub- 
tend equal  angles  at  their  respective  centres,  are  in  the  direct  proportion 
of  their  radii,  and  as  the  arc  N  A  is  greater  than  B  D,  and  that  again 
than  G  E,  it  follows  that  the  radius  N  x  must  be  greater  than  By,  and 
B^  than  Ez.  Thus  it  appears  that  the  mutual  intersections  of  the 
plumb-lines  will  not,  as  in  the  sphere,  all  coincide  in  one  point  C,  the 
centre,  but  will  be  arranged  along  a  certain  curve,  x  i/  z  (which  will  be 
rendered  more  evident  by  considering  a  number  of  intermediate  stations). 


126 


OUTLINES   OF  ABTRONOMT. 


If' !  I 


To  this  curve  geometers  hare  given  the  name  of  the  evolute  of  the  curve 
N  A  B  D  G  E,  from  whose  centres  of  curvature  it  is  constructed. 

(220.)  lu  the  flattening  of  a  round  figure  at  two  opposite  points,  and 
its  protuberance  at  points  rectangularly  situated  to  the  former,  we  recog- 
nize the  distinguishing  feature  of  the  elliptic  form.  Accordingly,  the 
next  and  simplest  supposition  that  we  can  make  respecting  the  nature  of 
the  meridian,  since  it  is  proved  not  to  be  a  circle,  is,  that  it  is  an  ellipse, 
or  nearly  so,  having  N  S,  the  axis  of  the  earth,  for  its  shorter,  and  E  F, 
the  equatorial  diameter,  for  its  longer  axis;  and  chat  the  form  of  the 
earth's  surface  is  that  which  would  arise  from  making  such  a  curve  revolve 
about  its  shorter  axis  N  S.  This  agrees  well  with  the  general  course  of 
the  increase  of  the  degree  in  going  from  the  equator  to  the  pole.  In  the 
ellipse,  the  radius  of  curvature  at  E,  the  extremity  of  the  longer  axis  is 
the  least,  and  at  that  of  the  shorter  axis,  the  greatest  it  admits,  and  the 
form  of  its  evolute  agrees  with  that  here  represented.'  Assuming,  thee, 
that  it  is  an  ellipse,  the  geometrical  properties  of  that  curve  enable  us  to 
assign  the  proportion  between  the  lengths  of  its  axes  which  shall  corre- 
spond to  any  proposed  rate  of  variation  in  its  curvature;  as  well  as  to  fix 
upon  their  absolute  lengths,  corresponding  to  any  assigned  length  of  the 
degree  in  a  given  latitude.  Without  troubling  the  reader  with  the  inves- 
tigation, (which  may  be  found  in  any  work  on  the  conic  sections,)  it  will 
be  sufficient  to  state  results  which  have  been  arrived  at  by  iV  most  sys- 
tematic combinations  of  the  measured  arcs  which  have  hitherto  been  made 
by  geometers.  The  most  recent  is  that  of  Bessel',  who  by  a  combination 
of  the  ten  arcs,  marked  B  in  our  table,  has  concluded  the  dimensions  of 
the  terrestrial  spheroid  to  be  as  follows : — 

Greater  or  equatorial  diameter         ... 
Lesser  or  polar  diameter         .... 
Difference  of  diameters,  or  polar  compression 
Proportion  of  diameters  as  299-15  to  298-15. 

The  other  combination  whose  results  we  shall  state,  is  that  of  Mr. 
Airy',  who  concludes  as  follows: 


Feet 

Miles. 

41,847,192  = 

7926-604 

41,707,324  = 

7899-114 

139,768  = 

26-471 

Equatorial  diameter         .... 

Polar  diameter 

Polar  compression  ..... 
Proportion  of  diameters  as  299-33  to  29833. 


Feet. 
=  41, 847,4  ^e 
=  41,707,620 
=        139,806 


MileR. 
7925-648 
7899-170 
26-478 


'  The  dotted  lines  are  the  portions  of  the  evolute  belonging  to  the  other  quadrants. 

*  Schumacher's  Astronomische  Nachrichten,  Nos.  333,  334,  335,  438. 

*  Encycloptedia  Metropolitana,  "  Figure  of  the  Earth"  (1831). 


DIMENSIONS  OF  THE   EARTH. 


12- 


These  oonolusions  are  bosod  on  the  considorotion  of  thoHO  13  aios,  to 
which  the  letter  A  is  annexed',  and  of  one  other  are  of  1°  7'  31"lj 
measured  in  Piedmont  by  Plana  and  Carliui,  whose  disoordanco  with  the 
rest,  owing  to  local  causes  hereafter  to  bo  explained,  arising  from  the  ex- 
ceedingly mountainous  nature  of  the  country,  render  the  propriety  of  so 
employing  it  very  doubtful.  Be  that  as  it  may,  the  strikingly  near  ac- 
cordance of  the  two  sets  of  dimensions  is  such  as  to  inspire  the  greatest 
confidence  in  both.  The  measurement  at  the  Cape  of  Good  Hope  by 
Laciiille,  also  used  in  this  determination,  has  always  been  regarded  as 
unsatisfactory,  and  has  recently  been  demonstrated  by  Mr.  Maclcar  to  be 
erroneous  to  a  considerable  extent.  The  omission  of  the  former,  and  the 
substitution  for  the  latter,  of  the  far  preferable  result  of  Mr.  Maclcar's 
second  measurement  would  induce,  however,  but  a  trifling  change  in  the 
final  result. 

(221.)  Thus  we  see  that  the  rough  diameter  of  8000  miles  wo  have 
hitherto  used,  is  rather  too  great,  the  excess  being  about  100  miles,  or 
g'gth  part.  As  convenient  numbers  to  remember,  the  reader  may  bear  in 
mind,  that  in  our  latitude  there  are  just  as  many  thousands  of  feet  in  a 
degree  of  the  meridian  as  there  are  days  in  the  year  (865)  :  that,  speak- 
ing loosely,  a  degree  is  about  70  British  statute  miles,  and  a  second  about 
100  foet ;  that  the  equatorial  circumference  of  the  earth  is  a  little  less 
than  25,000  miles  (24,899),  and  the  ellipticity  or  polar  flattening 
amounts  to  one  SOOth  part  of  the  diameter. 

(222.)  The  two  sets  of  results  above  stated  are  placed  in  juxtaposition, 
and  the  particulars  given  more  in  detail  than  may  at  first  sight  appear 
consonant,  either  with  the  general  plan  of  this  work,  or  the  state  of  the 
reader's  presumed  acquaintance  with  the  subject.  But  it  is  of  importance 
that  he  should  early  be  made  to  see  how,  in  astronomy,  results  in  admira- 
ble concordance  emerge  from  data  accumulated  from  totally  diflFerent  quar- 
ters, and  how  local  and  accidental  irregularities  in  the  data  themselves 
become  neutralized  and  obliterated  by  their  impartial  geometrical  treat- 
ment. In  the  cases  before  us,  the  modes  of  calculation  followed  are 
widely  different,  and  in  each  the  mass  of  figures  to  be  gone  through  to 
arrive  at  the  result,  enormous. 

(223.)  The  supposition  of  an  elliptic  form  of  the  earth's  section  through 
the  axis  is  recommended  by  its  simplicity,  and  confirmed  by  comparing 
the  numerical  results  we  have  just  set  down  with  those  of  actual  measure- 
ment. When  this  comparison  is  executed,  discordances,  it  is  true,  are 
observed,  which,  although  still  too  great  to  be  referred  to  error  of 

'  In  those  which  have  both  A  and  B,  the  numbers  used  by  Mr.  Airy  differ  slightlv 
from  Besaers,  which  are  those  we  have  preferred. 


t' 

n 

! 


1 

-■»■>' 

i:'§ 

w 

m 

n 

*  \ 


128 


OUTLINES   OF  ASTRONOMY. 


measurement,  arc  yet  no  nmall,  compared  to  the  errors  which  would  result 
from  the  spherical  hypothesis,  as  completely  to  justify  our  regarding  the 
earth  as  an  ellipsoid,  and  referring  the  observed  deviations  to  either  local 
or,  if  general,  to  comparatively  small  causes. 

(224.)  Now,  it  is  highly  satisfactory  to  find  that  the  general  elliptical 
figure  thus  practically  proved  to  exist,  is  precisely  what  ought  theoretically 
to  result  from  the  rotation  of  the  earth  on  its  axis.  For,  let  us  suppose 
the  earth  a  sphere,  at  rest,  of  uniform  materials  throughout,  and  exter- 
nally covered  with  an  ocean  of  equal  depth  in  every  part.  Under  such 
circumstances  it  would  obviously  be  in  a  state  of  equilibrium ;  and  the 
water  on  its  surface  would  have  no  tendency  to  run  one  way  or  the  other. 
Suppose,  now,  a  quantity  of  its  materials  were  token  from  the  polar 
regions,  and  piled  up  all  around  the  equator,  so  as  to  produce  that  dif- 
ference of  the  polar  and  equatorial  diameters  of  26  miles  which  we  know 
to  exist.  It  is  not  less  evident  that  a  mountain  ridge  or  equatorial  conti- 
nent,  onhjy  would  bo  thus  formed,  from  which  the  water  would  run  down 
the  excavated  part  at  the  poles.  However  solid  matter  might  rest  where 
it  was  placed,  the  liquid  part,  at  least,  would  not  remain  there,  any  more 
than  if  it  were  thrown  on  the  side  of  a  hill.  The  consequence  therefore. 
Would  bo  the  formation  of  two  great  polar  seas,  hemmed  in  all  round  by 
equatorial  land.  Now,  this  is  by  no  means  the  case  in  nature.  The 
ocean  occupies,  indi£ferently,  all  latitudes,  with  no  more  partiality  to  the 
polar  than  to  the  equ'itorial.  Since,  then,  as  we  see,  the  water  occupies 
an  elevation  above  the  centre  no  less  than  13  miles  greater  at  the  equator 
than  at  the  poles,  and  yet  manifests  no  tendency  to  leave  the  former  and 
run  towards  the  latter,  it  is  evident  that  it  must  be  retained  in  that 
situation  by  some  adequate  jaower.  No  such  power,  however,  would  exist 
in  the  case  we  have  supposed,  which  is  therefore  not  conformable  to 
nature.  In  other  words,  the  spherical  form  is  not  the  Jujure  of  equili- 
brium ;  and  there/ore  the  earth  is  either  not  at  rest,  or  is  so  internally 
constituted  as  to  attract  the  water  to  its  equatorial  regions,  and  retain  it 
there.  For  the  latter  supposition  there  is  no  primd  facie  probability,  nor 
any  analogy  to  lead  us  to  such  an  idea.  The  former  is  in  accordance  with 
all  the  phenomena  of  the  apparent  diurnal  motion  of  the  heavens;  and 
therefore,  if  it  will  furnish  us  with  the  power  in  question,  we  can  have  no 
hesitation  in  adopting  it  as  the  true  one. 

(225.)  Now,  every  b«"dy  knows  that  when  a  weight  is  whirled  round, 
it  acquires  thereby  a  tdudency  to  recede  from  the  centre  of  its  motion  j 
which  is  called  the  centrifugal  force.  A  stone  whirled  round  in  a  sling 
is  a  common  illustration ;  but  a  better,  for  our  present  purpose,  will  be  a 
pail  of  water,  suspended  by  a  cord,  and  mode  to  spin  rmmdy  while  the 


rUK  OF  THR    EARTH. 


129 


Fig.  88 


II      t  •*    f 


I      / 


cord  hangs  perpendicularly.  The  surface  of  the  water,  instead  of  re- 
maining horizontal,  will  become  concave,  as  in  the  figure.  The  centri- 
fugal force  generates  a  tendency  in  all  the  water  to  leave  the  axis,  and 
press  towards  the  circumference ;  it  is,  therefore,  urged  against  the  pail, 
and  forced  up  its  sides,  till  the  excess  of  height,  and  consequent  increase 
of  pressure  downwards,  just  counterbalances  its  centrifugal  force,  and  a 
state  of  equilibrium  is  attained.  The  experiment  is  a  very  easy  and 
instructive  one,  and  is  admirably  calculated  to  show  how  the  form  of 
equih'hrium  accommodates  itself  to  varying  circumstances.  If,  for  ex- 
ample, we  allow  the  rotation  to  cease  by  degrees,  as  it  becomes  slower  we 
shall  see  the  concavity  of  the  water  regularly  diminish;  the  elevated 
outward  portion  will  descend,  and  the  depressed  central  rise,  while  all  the 
time  a  perfectly  smooth  surface  is  maintained,  till  the  rotation  is  ex- 
hausted, when  the  water  resumes  its  horizontal  state. 

(226.)  Suppose,  then,  a  globe,  of  the  size  of  the  earth,  at  rest,  and 
covered  with  a  uniform  ocean,  were  to  be  set  in  rotation  about  a  certain 
axis,  at  first  very  slowly,  but  by  degrees  more  rapidly,  till  it  turned  round 
once  in  twenty-four  hours ;  a  centrifugal  force  would  be  thus  generated, 
whose  general  tendency  would  be  to  urge  the  water  at  every  point  of  tbo 
surface  to  recede  from  the  axis.  A  rotation  might,  indeed,  be  conceived 
so  swift  as  to  flirt  the  whole  ocean  from  the  surface,  like  water  from  a 
mop.  But  this  would  require  a  far  greater  velocity  than  what  we  now 
9 


lao 


OUTLINES   OF  ASTRONOMY. 


! 


speak  of.  In  the  case  supposed,  the  weight  of  the  water  would  still  keep 
it  on  the  earth ;  and  the  tendency  to  recede  from  the  axis  could  only  be 
satisfied,  therefore,  by  the  water  leaving  the  poles,  and  flowing  towards 
the  equator  j  there  heaping  itself  up  in  a  ridge,  just  as  the  water  in  our 
pail  accumulates  against  the  side ;  and  being  retained  in  opposition  to  its 
weight,  or  natural  tendency  towards  the  centre,  by  the  pressure  thus 
caused.  This,  however,  could  not  take  place  without  laying  dry  the 
polar  portions  of  the  land  in  the  form  of  immensely  protuberant  conti- 
nents ;  and  the  difference  of  our  supposed  cases,  therefore,  is  this  :  —  iu 
the  former,  a  great  equatorial  continent  and  polar  seas  would  be  formed ; 
in  the  latter,  protuberant  land  would  appear  at  the  poles,  and  a  zone  of 
ocean  be  disposed  around  the  equator.  This  would  be  the  first  or  im- 
mediate effect.  Let  us  now  see  what  would  afterwards  happen,  in  the 
two  cases,  if  things  were  allowed  to  take  their  natural  course. 

(227.)  The  sea  is  constantly  beating  on  the  land,  grinding  it  down, 
and  scattering  its  worn-off  particles  and  fragments,  in  the  state  of  mud 
and  pebbles,  over  its  bed.  Geological  facts  afford  abundant  proof  that  the 
existing  continents  have  all  of  them  undergone  this  process,  even  more 
than  once,  and  been  entirely  torn  in  fragments,  or  reduced  to  powder,  and 
submerged  and  reconstructed.  Land,  in  this  view  of  the  subject,  loses 
its  attribute  of  fixity.  As  a  mass  it  might  hold  together  in  opposition  to 
forces  which  the  water  freely  obeys;  but  in  its  state  of  successive  or 
simultaneous  degradation,  when  disseminated  through  the  water,  in  the 
state  of  sand  or  mud,  it  is  subject  to  all  the  impulses  of  that  fluid.  In 
the  lapse  of  time,  then,  the  protuberant  land  in  both  cases  would  be  des- 
troyed, and  spread  over  the  bottom  of  the  ocean,  filling  up  the  lower  parts, 
and  tending  continually  to  remodel  the  surface  of  the  solid  nucleus,  in  cor- 
respondence with  the  form  of  equilibrium  in  both  cases.  Thus,  after  a 
sufficient  lapse  of  time,  in  the  case  of  an  earth  at  rest,  the  equatorial  con- 
tinent, thus  forcibly  constructed,  would  again  be  levelled  and  transferred 
to  the  polar  excavations,  and  the  spherical  figure  be  so  at  length  restored. 
In  that  of  an  earth  in  rotation,  the  polar  protuberances  would  gradually 
be  cut  down  and  disappear,  being  transferred  to  the  equator  (as  being 
tJieH  the  deepest  sea),  till  the  earth  would  assume  by  degrees  the  form  we 
observe  it  to  have  —  that  of  a  flattened  or  oblate  ellipsoid. 

(228.)  We  ;  re  far  from  meaning  here  to  trace  the  process  %  which 
the  earth  really  assumed  its  actual  form ;  all  we  intend  is,  to  show  that 
this  is  the  form  to  which,  under  the  conditions  of  a  rotation  on  its  axis, 
it  must  tend ;  and  which  it  would  attain,  even  if  originally  and  (so  to 
upeak)  perversely  constituted  otherwise. 

(229.)  But,  further,  the  dimensions  of  the  earth  and  the  time  of  its 


VARIATION   OF  TERRESTRIAL  GRAVITY. 


131 


rotation  being  known,  it  is  easy  thence  to  calculate  the  exact  amount  of 
the  centrifugal  force/  which,  at  the  equator,  appears  to  be  ^g^th  part  of 
the  force  or  weight  by  which  all  bodies,  whether  solid  or  liquid,  tend  to 
fall  towards  the  earth.  By  this  fraction  of  its  weight,  then,  the  sea  at 
the  equator  is  lightened,  and  thereby  rendered  susceptible  of  being  sup- 
ported on  a  higher  level,  or  more  remote  from  the  the  centre  than  at  the 
poles,  where  no  such  counteracting  force  exists;  and  where,  in  conse- 
quence, the  water  may  be  considered  as  specifically  heavier.  Taking  this 
principle  as  a  guide,  and  combining  it  with  the  laws  of  gravity  (as  devel- 
oped by  Newton,  and  as  hereafter  to  be  more  fully  explained),  mathemati- 
cians have  been  enabled  to  investigate,  d  priori,  what  would  be  the  figure 
of  equilibrium  of  such  a  body,  constituted  internally  as  we  have  reason 
to  believe  the  earth  to  be ;  covered  wholly  or  partially  with  a  fluid ;  and 
revolving  uniformly  in  twenty-four  hours ;  and  the  result  of  this  inquiry 
is  found  to  agree  very  satisfactorily  with  what  experience  shows  to  be  the 
case.  From  their  investigations  it  appears  that  the  form  of  equilibrium 
is,  in  fact,  no  other  than  an  oblate  ellipsoid,  of  a  degree  of  ellipticity  very 
nearly  identical  with  what  is  observed,  and  which  would  be  no  doubt 
accurately  so,  did  we  know,  with  precision,  the  internal  constitution  and 
materials  of  the  earth. 

(230.)  The  confirmation  thus  incidentally  furnished,  of  the  hypothesis 
of  the  earth's  rotation  on  its  axis,  cannot  fail  to  strike  the  reader.  A 
deviation  of  its  figure  from  that  of  a  sphere  was  not  contemplated  among 
the  original  reasons  for  adopting  that  hypothesis,  which  was  assumed 
solely  on  account  of  the  easy  explanation  it  offers  of  the  apparent  diurnal 
motion  of  the  heavens.  Yet  we  see  that,  once  admitted,  it  draws  with  it, 
as  a  necessary  consequence,  this  other  remarkable  phenomenon,  of  which 
no  other  satisfactory  account  could  be  rendered.  Indeed,  so  direct  is  their 
connection,  that  the  ellipticity  of  the  earth's  figure  was  discovered  and 
demonstrated  by  Newton  to  be  a  consequence  of  its  rotation,  and  its  amount 
actually  calculated  by  him,  long  before  any  measurement  had  suggested 
such  a  conclusion.  As  we  advance  with  our  subject,  we  shall  find 
the  same  simple  principle  branching  out  into  a  whole  train  of  singular 
and  important  consequences,  some  obvious  enough,  others  which  at  first 
seem  entirely  unconnected  with  it,  and  which,  until  traced  by  Newton  up 
to  this  their  origin,  had  ranked  among  the  most  inscrutable  arcana  of 
astronomy,  as  well  as  among  its  grandest  phenomena. 

(231.)  Of  its  more  obvious  consequences,  we  may  here  mention  one 
which  falls  naturally  within  our  present  subject.      If  the  earth  really 


I* 

m  f 


Newton's  Principia,  iii.    Prop,  19. 


182 


OUTLU    iS  OP  ASTRONOMY. 


revoIvR  on  its  axis,  this  rotation  must  generate  a  centrifugal  force  (see 
art.  225,)  the  effect  of  which  must  of  course  be  to  counteract  a  certain 
portion  of  the  weight  of  every  body  situated  at  the  equator,  as  compared 
with  its  weight  at  the  poles,  or  in  any  intermediate  latitudes.  Now,  this 
is  fully  confirmed  by  experience.  There  is  actually  observed  to  exist  a 
difference  in  the  gravity,  or  downward  tendency,  of  one  and  the  same 
body,  when  conveyed  successively  to  stations  in  different  latitudes.  Ex- 
periments made  with  the  greatest  care,  and  in  every  accessible  part  of  the 
globe,  have  fully  demonstrated  the  fact  of  a  regular  and  progressive 
increase  in  the  weights  of  bodies  corresponding  to  the  increase  of  lati- 
tude, and  fixed  its  amount  and  the  law  of  its  progression.  From  these  it 
appears,  that  the  extreme  amount  of  this  variation  of  gravity,  or  the 
difference  between  the  equatorial  and  polar  weights  of  one  and  the  same 
mass  of  matter,  is  1  part  in  194  of  its  whole  weight,  the  rate  of  increase 
in  travelling  from  the  equator  to  the  pole  being  as  the  square  of  the  sine 
of  the  latitude. 

(232.)  The  reader  will  here  naturally  inquire,  what  is  meant  by 
speaking  of  the  same  body  as  having  different  weights  at  different  sta- 
tions; and,  how  such  a  fact,  if  true,  can  be  ascertained.  When  we 
weigh  a  body  by  a  balance  or  a  steelyard,  we  do  but  counteract  its  weight 
by  the  equal  weight  of  another  body  under  the  very  same  circumstances ; 
and  if  both  the  body  weighed  and  its  counterpoise  be  removed  to  another 
station,  their  gravity,  if  changed  at  all,  will  be  changed  equally,  so  that 
they  will  still  continue  to  counterbalance  each  other.  A  difference  in  the 
intensity  of  gravity  could,  therefore,  never  be  detected  by  these  means ; 
nor  is  it  in  this  sense  that  we  assert  that  a  body  weighing  194  pounds  at 
the  equator  will  weigh  195  at  the  pole.  If  counterbalanced  in  a  scale 
or  steelyard  at  the  former  station,  an  additional  pound  placed  in  one  or 
other  scale  at  the  latter  would  inevitably  sink  the  beam. 

(233.)  The  meaning  of  the  proposition  may  be  thus  explained:  — 
Conceive  a  weight  x  suspended  at  the  equator  by  a  string  without  weight 


Fig.  85. 


VARIATION   OF  TERRESTRIAL  aRAVITY. 


133 


passing  over  a  pulley,  A,  and  conducted  (supposing  such  a  thing  possi- 
ble) over  other  pulleys,  such  as  B,  round  the  earth's  convexity,  till  the 
other  end  hung  down  at  the  pole,  and  there  sustained  the  weight  y.  If, 
then,  the  weights  x  and  y  ^ere  such  as,  at  any  one  station,  equatorial  or 
polar,  would  exactly  counterpoise  each  other  on  a  balance,  or  when  sus- 
pended side  by  side  over  a  single  pulley,  they  would  not  counterbalance 
each  other  in  this  supposed  situation,  but  the  polar  weight  y  would  pre- 
ponderate ;  and  to  restore  the  equipoise  the  weight  x  must  be  increased 
by  y^,jth  part  of  its  quantity. 

(234.)  The  means  by  which  this  variation  of  gravity  may  be  shown 
to  exist,  and  its  amount  measured,  are  twofold  (like  all  estimations  of 
mechanical  power,)  statical  and  dynamical.  The  former  consists  in 
putting  the  gravity  of  a  weight  in  equilibrium,  not  with  that,  of  another 
weight,  but  with  a  natural  power  of  a  different  kind  not  liable  to  be 
affected  by  local  situation.  Such  a  power  is  the  elastic  force  of  a  spring. 
Let  A  B  C  be  a  strong  support  of  brass  standing  on  the  foot  A  E  D  cast 
in  one  piece  with  it,  into  which  is  let  a  smooth  plate  of  agate,  D,  which 
can  be  adjusted  to  perfect  horizontality  by  a  level.     At  C  let  a  spiral 


spring  G  be  attached,  which  carries  at  its  lower  end  a  weight  F,  polished 
and  convex  below.  The  length  and  strength  of  the  spring  must  be  so 
adjusted  that  the  weight  F  shall  be  sustained  by  it  just  to  swing  clear  of 
contact  with  the  agate  plate  in  the  highest  latitude  at  which  it  is  intended 
to  use  the  instrument.  Then,  if  smoU  weights  be  added  cautiously,  it 
may  be  made  to  descend  till  it  just  grazes  the  agate,  a  contact  which  can 
be  made  with  the  utmost  imaginable  delicacy.     Let  these  weights  bo 


134 


OUTLINES   OF  ASTRONOMT. 


M 


WW 


It: 


noted ;  the  weight  F  detached ;  the  spring  G  carefully  lifted  off  its  hook, 
and  secured,  for  travelling,  from  rust,  strain,  or  disturbance,  and  the 
whole  apparatus  conveyed  to  a  station  in  a  lower  latitude.  It  will  then 
be  found,  on  remounting  it,  that,  although  loaded  with  the  same  addi- 
tional weights  as  before,  the  weight  F  will  no  longer  have  power  enough 
to  stretch  the  spring  to  the  extent  required  for  producing  a  similar  con- 
tact. More  weights  will  require  to  be  added ;  and  the  additional  quan- 
tity necessary  will,  it  is  evident,  measure  the  difference  of  gravity 
between  the  two  stations,  as  exerted  on  the  whole  qiiantity  of  pendent 
mattei',  i.  e.  the  sum  of  the  weight  F  and  half  that  of  the  spiral  spring 
itself.  Granting  that  a  spiral  spring  can  be  constructed  of  such  strength 
and  dimensions  that  a  weight  of  10,000  grains,  including  its  own,  shall 
produce  an  elongation  of  10  inches  without  permanently  straining  it,' 
one  additional  grain  will  produce  a  further  extension  of  yu^u^th  of  an 
inch,  a  quantity  which  cannot  possibly  be  mistaken  in  such  a  contact  as 
that  in  question.  Thus  we  should  be  provided  with  the  means  of  mea- 
suring the  power  of  gravity  at  any  station  to  within  xuiiia*^  ^^  ^*^  whole 
quantity. 

(235.)  The  other,  or  dynamical  process,  by  which  the  force  urging  any 
given  weight  to  tho  earth  may  be  determined,  consists  in  ascertaining  the 
velocity  imparted  by  it  to  the  weight  when  suffered  to  fall  freely  in  a  given 
time,  as  one  second.  This  velocity  cannot,  indeed,  be  directly  measured ; 
but  indirectly,  the  principles  of  mechanics  furnish  an  easy  and  certain 
means  of  deducing  it,  and,  consequently,  the  intensity  of  gravity,  by  ob- 
serving the  oscillations  of  a  pendulum.  It  is  proved  from  mechanical 
principles'',  that,  if  one  and  the  same  pendulum  be  made  to  oscillate  at 
different  stations,  or  under  the  influence  of  different  forces,  and  the 
numbers  of  oscillations  made  in  the  same  time  in  each  case  be  counted, 
the  intensities  of  the  forces  will  be  to  each  other  as  the  squares  of  the 
numbers  of  oscillations  made,  and  thus  their  proportion  becomes  known. 
For  instance,  it  is  found  that,  under  the  equator,  a  pendulum  of  a  certain 
form  and  length  makes  86,400  vibrations  in  a  mean  solar  day ;  and  that, 
when  transported  to  London,  the  same  pendulum  makes  86,535  vibrations 

'  Whether  the  process  above  described  could  ever  be  so  far  perfected  and  refined  as 
to  become  a  substitute  for  the  use  of  the  pendulum  must  depend  on  the  decree  of 
permanence  and  uniformity  of  action  of  springs,  on  the  constancy  or  variability  of  the 
effect  of  temperature  on  their  clastic  force,  on  the  possibility  of  transporting  them, 
absolutely  unaltered,  from  p'ace  to  place,  &c.  The  great  advantages,  however, 
which  such  an  apparatus  and  mode  of  observation  would  possess,  i.i  point  of  conve- 
nience, cheapness,  porlobility,  and  expedition,  over  the  preb^nt  laborious,  tedious,  ai  I 
exuensive  process,  render  the  attempt  well  worth  making. 
Newton's  Principia,  ii.  Prop.  24,  Cor.  3. 


GRAVITY   OF   A   SPHEROID 


135 


in  tlio  same  time.  Hence  we  conclude,  that  the  inttinsity  of  the  force 
urging  the  pendulum  downwards  at  the  equator  is  to  that  at  London  as 
(86,400)*  to  (86,535)',  or  as  1  to  1-00315;  or,  in  other  words,  that  a 
mass  of  matter  weighing  in  London  100,000  pounds,  exerts  the  same 
pressure  on  the  ground,  or  the  same  eflFort  to  crush  a  body  placed  below 
it,  that  100,315  of  the  same  pounds  transported  to  the  equator  would 
exert  there. 

(236.)  Experiments  of  this  kind  have  been  made,  as  above  stated,  with 
the  utmost  care  and  minutest  precaution  to  ensure  exactness  In  all  acces- 
sible latitudes;  and  their  general  and  final  result  has  been,  to  give  j\j 
for  the  fraction  expressing  the  difference  of  gravity  at  the  equator  and 
poles.  Now,  it  will  not  fail  to  be  noticed  by  the  reader,  and  will,  pro- 
bably, occur  to  him  as  an  objection  against  the  explanation  here  given  of 
the  fact  by  the  earth's  rotation,  that  this  differs  materially  from  the  frac- 
tion jI^  expressing  the  centrifugal  force  at  the  equator.  The  difference 
by  which  the  former  fraction  exceeds  the  latter  is  ^^^,  a  small  quantity 
in  itself,  but  still  far  too  large,  compared  with  the  others  in  question,  not 
to  be  distinctly  accounted  for,  and  not  to  prove  fatal  to  this  explanation 
if  it  will  not  render  a  strict  account  of  it. 

(237.)  The  mode  in  which  this  difference  arises  affords  a  curious  and 
instructive  example  of  the  indirect  influence  which  mechanical  causes 
often  exercise,  and  of  which  astronomy  furni&Los  innumerable  instances. 
The  rotation  of  the  earth  gives  rise  to  the  centrifugal  force ;  the  centri- 
fugal force  produces  an  ellipticity  in  the  form  of  the  earth  itself;  and  this 
very  ellipticity  of  form  modifies  its  power  of  attraction  on  bodies  placed 
at  its  surface,  and  thus  gives  rise  to  the  difference  in  question.  Here, 
then,  we  have  the  same  cause  exercising  at  once  a  direct  and  an  indirect 
influence.  The  amount  of  the  former  is  easily  calculated,  that  of  the 
latter  with  far  more  difficulty,  by  an  intricate  and  profound  application  of 
geometry,  whose  steps  we  cannot  pretend  to  irace  in  a  work  like  the  pre- 
sent, and  can  only  state  its  nature  and  result. 

(238.)  The  weight  of  a  body  (cousidered  as  undiminished  by  a  centri- 
fugal force)  is  the  effect  of  the  earth's  attraction  on  it.  This  attraction, 
as  Newton  has  demonstrate(^;  consists,  not  in  a  tendency  of  all  matter  to 
any  one  particular  centre,  buo  in  a  disposition  of  every  particle  of  matter 
in  the  universe  to  press  towards,  and  if  not  opposed  to  approach  to,  every 
other.  The  attraction  of  the  earth,  then,  on  a  body  placed  on  its  surface, 
is  not  a  simple  but  a  complex  force,  resulting  from  the  separate  attractions 
of  all  its  parts.  Now,  it  is  evident,  that  if  the  earth  were  a  perfect  sphere, 
the  attraction  exerted  by  it  on  a  body  any  where  placed  on  its  surface, 
whether  at  its  equator  or  pole,  must  be  exactly  alike, — for  the  simple 


136 


OUTLINES   OF  ASTRONOMY. 


,'1 1) 


reason  of  the  exact  symmetry  of  the  sphere  in  every  direction.  It  is  net 
less  evident  that,  the  earth  being  elliptical,  and  this  symmetry  or  simili- 
tude of  all  its  parts  not  existing,  the  same  result  cannot  be  expected.  A 
body  placed  at  the  equator,  and  a  similar  one  at  the  pole  of  a  flattened 
ellipsoid,  stand  in  a  different  geometrical  relation  to  the  mass  as  a  whole. 
This  difference,  without  entering  further  into  particulars,  may  be  expected 
to  draw  with  it  a  difference  in  its  forces  of  attraction  on  the  two  bodies. 
Calculation  confirms  this  idea.  It  is  a  question  of  purely  mathematical 
investigation,  and  has  been  treated  with  perfect  clearness  and  precision 
by  Newton,  Maclaurin,  Clairaut,  and  many  other  eminent  geometers ;  and 
the  result  of  their  investigations  is  to  show  that,  owing  to  the  elliptic  form 
of  the  earth  alone,  and  independent  of  the  centrifugal  force,  its  attraction 
ought  to  increase  the  weight  of  a  body  in  goiug  from  the  equator  to  the 
pole  by  almost  exactly  ^^x;^^  P^^^  f  ^^ich,  together  with  i;|^th  due  to 
the  centrifugal  force,  make  up  the  whole  quantity,  y^^th,  observed. 

(239.)  Another  great  geographical  phenomenon,  which  owes  its  exis- 
tence to  the  earth's  rotation,  is  that  of  the  trade-winds.  These  mighty 
currents  in  our  atmosphere,  on  which  so  important  a  part  of  navigation 
depends,  arise  from,  1st,  the  unequal  exposure  of  the  earth's  surface  to 
the  sun's  rays,  by  which  it  is  unequally  heated  in  different  latitudes; 
and,  2dly,  from  that  general  law  in  the  constitution  of  all  fluids,  in  virtue 
of  which  they  occupy  a  larger  bulk,  and  become  specifically  lighter  when 
hot  than  when  cold.  These  causes,  combined  with  the  earth's  rotation 
from  west  to  east,  afford  an  easy  and  satisfactory  explanation  of  the  mag- 
nificent phenomena  in  question. 

(240.)  It  is  a  matter  of  observed  fact,  of  which  we  shall  give  the 
explanation  farther  on,  that  the  sun  is  constantly  vertical  over  some  one 
or  other  part  of  the  earth  between  two  parallels  of  latitude,  called  the 
tropics,  respectively  23 i°  north,  and  as  much  south  of  the  equator;  and 
that  the  whole  of  that  zone  or  belt  of  the  earth's  surface  included  between 
the  tropics,  and  equally  divided  by  the  equator,  is,  in  consequence  of  the 
.;reat  altitude  attained  by  the  sun  in  its  diurnal  course,  maintained  at  a 
much  higher  temperature  than  those  regions  to  the  north  and  south 
which  lie  nearer  the  poles.  Now,  the  heat  thus  acquired  by  the  earth's 
surface  is  communicated  to  the  incumbent  air,  which  is  thereby  expanded, 
and  rendered  specifically  lighter  than  the  air  incumbent  on  the  rest  of  the 
globe.  It  is  therefore,  in  obedience  to  the  general  laws  of  hydrostatics, 
displaced  and  buoyed  up  from  the  surface,  and  its  place  occupied  by 
colder,  and  therefore  heavier  air,  which  glides  in,  on  both  sides,  along  the 
surface,  from  the  regions  beyond  the  tropics ;  while  the  displaced  air,  thus 


TRADE   WINDS. 


ISI 


raised  above  its  due  level,  and  unsustained  by  a-/  lateral  pressure,  flows 
over,  as  it  were,  and  forms  an  upper  current  in  the  contrary  direction,  o: 
towards  the  poles ;  which,  being  cooled  in  its  course,  and  also  sucked 
down  to  supply  the  deficiency  in  the  extra-tropical  regions,  keeps  up  thus 
a  continual  circulation. 

(241.)  Since  the  earth  revolves  about  an  axis  passing  through  the 
poles,  the  equatorial  portion  of  its  surface  has  the  greatest  velocity  of 
rotation,  and  all  other  parts  less  in  the  proportion  of  the  radii  of  the 
circles  of  latitude  to  which  they  correspond.  But  as  the  air,  when  rela- 
tively and  appareiifly  at  rest  on  any  part  of  the  earth's  surface,  is  only  so 
because  in  reality  it  participates  in  the  motion  of  rotation  proper  to  that 
part,  it  follows  that  when  a  mass  of  air  near  the  poles  is  transferred  to 
the  region  near  the  equator  by  any  impulse  urging  it  directly  towards  that 
circle,  in  every  point  of  its  progress  towards  its  new  situation  it  must  be 
found  deficient  in  rotatory  velocity,  and  therefore  unable  to  keep  up  with 
the  speed  of  the  new  surface  over  which  it  is  brought.  Hence,  the  cur- 
rents of  air  which  set  in  towards  the  equator  from  the  north  snd  south 
must,  as  they  glide  aL.  g  the  surface,  at  the  same  time  lag,  or  hang  back, 
and  drag  upon  it  in  the  direction  opposite  to  the  earth's  rotation,  i.  e. 
from  east  to  west.  Thus  these  currents  which  but  for  the  rotation  would 
be  simply  northerly  and  southerly  winds,  acquire,  from  this  cause,  a  rela- 
tive direction  towards  the  west,  and  assume  the  character  of  permanent 
north-easterly  and  south-easterly  winds. 

(242.)  Were  any  considerable  mass  of  air  to  be  suddenly  transferred 
from  beyond  the  topics  to  the  equsttor,  the  difference  of  the  rotatory  velo- 
cities proper  to  the  two  situations  would  be  so  great  as  to  produce  not 
merely  a  wind,  but  a  tempest  of  the  most  destructive  violence.  But  this 
is  not  the  case:  tho  advance  of  the  air  from  the  north  and  south  is 
gradual,  and  all  the  while  the  earth  is  condnually  acting  on,  and  by  the 
friction  of  its  surface  accelerating  its  rotatory  velocity.  Supposing  its 
progress  towards  the  equator  to  cease  at  any  point,  this  cause  would 
almost  immediately  communicate  to  it  the  deficient  motion  of  rotation, 
after  which  it  would  revolve  quietly  with  the  earth,  and  be  at  relative 
rest.  We  have  only  to  call  to  mind  the  comparative  thinness  of  the  coat- 
ing which  the  atmosphere  forms  around  the  globe  (art.  35),  and  the  im- 
mense mass  of  the  latter,  compared  with  th.  former  (which  it  exceeds  at 
least  100,000,000  times),  to  appreciate  fully  the  absolute  command  of 
any  extensive  territory  of  the  earth  over  the  atmosphere  immediately 
incumbent  on  it,  in  point  of  motion. 

(243.)  It  follows  from  this,  then,  that  as  the  winds  on  botli  sides  ap- 


138 


OUTLINES   OF  ASTRONOMY. 


prouch  the  equator,  their  easterly  tendency  must  diminisb.'  The  lengths 
of  the  diurnal  circles  increase  very  slowly  in  the  immediate  vicinity 
of  the  equator,  and  for  several  degrees  on  either  side  of  it  hardly  change 
at  all.  Thus  the  friction  of  the  surface  has  more  time  to  act  in  accelera- 
ting the  velocity  of  the  air,  bringing  it  towards  a  state  of  relative  rest, 
and  diminishing  thereby  the  relative  set  of  the  currents  from  east  to  west, 
which,  on  the  other  band,  ia  feebly,  and,  at  length,  not  at  all  reinforced 
by  the  cause  which  originally  produced  it.  Arrived,  then,  at  the  equator, 
the  trades  must  be  expected  to  lose  their  easterly  character  altogether. 
But  not  only  this  but  the  northern  and  southern  currents  here  meeting 
and  opposing,  will  mutually  destroy  eacli  other,  leaving  only  such  pre- 
ponderancy  as  may  be  due  to  a  difference  of  local  causes  acting  in  the  two 
hemispheres,  —  which  in  some  regions  around  the  equator  may  lie  one 
way,  in  some  another. 

(244.)  The  result,  tiien,  must  be  the  production  of  two  great  tropical 
belts,  in  the  northern  of  which  a  constant  north-«asterly,  and  in  the 
southern  a  south-easterly,  wind  must  prevail,  while  the  winds  in  the 
equatorial  belt,  which  separates  the  two  former,  should  be  comparatively 
calm  and  free  from  any  steady  prevalence  of  easterly  character.  All 
these  consequences  are  agreeable  to  observed  fact,  and  the  system  of  aerial 
currents  above  described  constitutes  in  reality  what  is  understood  by  the 
regular  trade  winds. 

(245.)  The  constant  friction  thus  produced  between  the  earth  and  at- 
mosphere in  the  regions  near  the  equator  must  (it  may  be  objected)  by 
degrees  reduce  and  at  length  destroy  the  rotation  of  the  whole  mass. 
The  laws  of  dynamics,  however,  render  such  a  consequence,  generally, 
impossible ;  and  it  is  easy  to  see,  in  the  present  case,  where  and  how  the 
compensation  takes  place.  The  heated  equatorial  air,  while  it  rises  and 
flows  over  towards  the  poles,  carries  with  it  the  rotatory  velocity  due  to 
its  equatorial  situation  into  a  higher  latitude,  where  the  earth's  surface 
has  less  motion.  Hence,  as  it  travels  northward  or  southward,  it  will 
gain  continually  more  and  more  on  the  surface  of  the  earth  in  its  diurnal 
motion,  and  assume  constantly  more  and  more  a  westerly  relative  direc- 
tion; and  when  at  length  it  returns  to  the  surface,  in  its  circulation, 
which  it  must  do  more  or  less  in  all  the  interval  between  the  tropics  and 
the  poles,  it  will  act  on  it  by  its  friction  as  a  powerful  south-west  wind  io 
the  northern  hemisphere,  and  a  north-west  in  the  southern,  ^^nd  restore  to 
it  the  impulse  taken  up  from  it  at  the  equator.     "We  have  here  the  origin 

•  See  Captain  Hall's  "  Fragments  of  Voyages  and  Travels,"  2d  series,  vol.  i.  p. 
162,  where  this  is  very  distinctly,  and,  so  far  as  I  am  aware,  for  the  first  time,  reasoned 


DETERMINATION   OF   LATITUDES, 


139 


of  the  south-west  and  westerly  gales  m  prevalent  in  our  latitudes,  and  of 
the  almost  universal  westerly  winds  in  tho  North  Atlantic,  which  are,  in 
fact,  nothing  else  than  a  part  of  the  geh.iixi  system  of  the  re-action  of  the 
trades,  and  of  the  process  by  which  the  equilibrium  of  the  earth's  mo- 
tion is  maintained  under  their  action.' 

(246.)  In  order  to  construct  a  map  or  model  of  the  earth,  and  obtain 
a  knowledge  of  the  distribution  of  sea  and  land  over  its  surface,  the  forms 
of  the  outlines  of  its  continents  and  islands,  the  courses  of  its  rivers  and 
mountain  chains,  and  the  relative  situations,  with  respect  to  each  other, 
of  those  points  which  chiefly  interest  us,  as  centres  of  human  habitation, 
or  from  other  causes,  it  is  necessary  to  possess  the  means  of  determining 
correctly  the  situation  of  any  proposed  station  on  its  surface.  For  this 
two  elements  require  to  be  known,  the  latitude  and  longitude,  the  former 
assigning  its  distance  from  the  poles  or  the  equator,  the  latter,  the  meri- 
dian on  which  that  distance  is  to  be  reckoned.  To  these,  in  strictness, 
should  be  ad^^ed,  its  height  above  the  sea  level ;  but  the  consideration  of 
this  had  better  be  deferred,  to  avoid  complicating  the  subject. 

(247.)  The  latitude  of  a  station  on  a  sphere  would  be  merely  the 
length  of  an  arc  of  the  meridian,  intercepted  between  the  station  and  the 
nearest  point  of  the  equator,  reduced  into  degrees.  (See  art.  88.)  But 
as  the  earth  is  elliptic,  this  mode  of  conceiving  latitudes  becomes  inappli- 
cable, and  we  are  compelled  to  resort  for  our  definition  of  latitude  to  a 
generalization  of  that  property  (art.  119,)  which  affords  the  readiest 
means  of  determining  it  by  observation,  and  which  has  the  advantage  of 
being  independent  of  the  figure  of  the  earth,  which,  after  all,  's  not 
exactly  an  ellipsoid,  or  any  known  geometrical  solid.  The  latitude  of  a 
station,  then,  is  the  altitude  of  the  elevated  pole,  and  is,  therefore,  astro- 
nomically determined  by  those  methods  already  explained  for  ascertaining 


:  ri- 
ll 


'  As  it  is  our  object  merely  to  illustrate  the  mode  in  which  the  earth's  rotation  affects 
the  atmosphere  on  the  great  scale,  we  onri^  all  consideration  of  local  periodical  winds, 
such  as  monsoons,  &c. 

It  seems  worth  inquiry,  whether  hurricanes  in  tropical  climates  may  not  arise  from 
portions  of  the  upper  currents  prematurely  diverted  downwards  before  their  relative 
velocity  has  been  sufficiently  reduced  by  friction  on,  and  gradual  mixing  with,  the 
lower  strata ;  and  so  dashing  upon  the  earth  with  that  tremendous  velocity  which  gives 
them  their  destructive  character,  and  of  which  hardly  any  rational  account  has  yet 
been  given.  But  it  by  no  means  follows  that  this  must  always  be  the  cacie.  In 
general,  a  rapid  transfer,  either  way,  in  latitude,  of  any  mass  of  air  which  Iccal  or 
temporary  causes  might  carry  ahove  the  immediate  reach  of  the  friction  of  the  eaHh^a 
surface,  would  give  a  fearful  exaggeration  to  its  velocity.  Wherever  such  a  mass 
should  strike  the  earth,  a  hurricane  might  arise  ;  and  should  two  such  masses  cncoun 
ter  in  mid  air,  a  tornado  of  any  degree  of  intensity  on  record  might  easily  resuit  from 
their  combination. 


140 


OUTLINES   OF  ASTRONOMY. 


that  important  clement.  In  consequence,  it  will  be  remembered  that,  to 
make  a  perfectly  correct  map  of  the  whole,  or  any  part  of  the  earth's 
surface,  equal  differences  of  latitude  are  not  represented  by  exactly  equal 
intervals  of  surface. 

(248.)  For  the  purposes  of  geodesical '  measurements  and  trigonome- 
trical surveys,  an  exceedingly  correct  determination  of  the  latitudes  of  the 
most  important  stations  is  required.  For  this  purpose,  therefore,  the 
zenith  sector  (an  instrument  capable  of  great  precision)  is  most  commonly 
used  to  observe  stars  passing  the  meridian  near  the  zenith,  whose  declina- 
tions have  become  known  by  previous  long  series  of  observations  at  fixed 
observatories,  and  which  are  therefore  called  standard  or  fundamental 
stars,     llecently  a  method*  has  been  employed  with  great  success,  which 


consists  in  the  use  of  an  instrument  similar  in  every  respect  to  the  transit 
instrument,  but  having  the  plane  of  motion  of  the  telescope  not  coinci- 
dent with  the  meridian,  but  with  the  prime  vertical,  so  that  its  axis  of 
rotation  prolonged  passes  through  the  north  and  south  points  of  the 
horizon.  Let  A  B  C  D  be  the  celestial  hemisphere  projected  on  the 
horizon,  P  the  pole,  Z  the  zenith,  A  B  the  meridian,  C  D  the  prime 
vertical,  Q  R  S  part  of  the  diurnal  circle  of  a  star  passing  near  the 
zenith,  whose  polar  distance  P  R  is  but  little  greater  than  the  co-latitude 
of  the  place,  or  the  arc  P  Z,  between  the  zenith  and  pole  (art.  112.) 
Then  the  moments  of  this  star's  arrival  on  the  prime  vertical  at  Q  and  S 

'  Tti,  the  earth ;  han  (from  itn,  to  bind,)  a  joining  or  connection  (of  parts.) 
*  Devised  originally  by  Romer.     Revived  cr  re-invented  by  BessaX.—Astr.  Nachr. 
No.  40. 


DETERMINATION  OF  LATITUDES. 


141 


will,  if  '^10  instrument  be  correctly  adjusted,  be  those  of  its  crossing  the 
middle  wire  in  the  field  of  view  of  the  telescope  (art.  160.)  Conse- 
quently the  interval  between  these  moments  will  be  the  time  of  the  star 
passing  from  Q  to  S,  or  the  measure  of  the  diurnal  arc  Q  R  S,  which 
corresponds  to  the  angle  Q  P  S  at  the  pole.  This  angle,  therefore,  be- 
comes known  hy  the  mere  observation  of  an  interval  of  time,  in  which  it 
is  not  even  necessary  to  know  the  error  of  the  clock,  and  in  which,  when 
the  star  passes  near  the  zenith,  so  that  the  interval  in  question  is  small, 
even  the  rate  of  the  clock,  or  its  gain  or  loss  on  true  sidereal  time,  may 
be  neglected.  Now  the  angle  Q  P  S,  or  its  half  Q  P  R,  and  P  Q  tho 
polar  distance  of  the  star,  being  known,  P  Z  the  zenith  distance  of  the 
polo  can  be  calculated  by  the  resolution  of  the  right-angled  spherical 
triangle  P  Z  Q,  and  thus  the  co-latitude  (and  of  course  the  latitude)  of 
the  place  of  observation  becomes  known.  The  advantages  gained  by  this 
mode  of  observation  are,  1st,  that  no  readings  of  a  divided  arc  are  needed, 
so  that  errors  of  graduation  and  reading  are  avoided:  2dly,  that  the 
arc  Q  R  S  is  very  much  greater  than  its  versed  sine  R  Z,  so  that  the 
difference  R  Z  between  the  latitude  of  the  place  and  the  declination  of 
the  star  is  given  1/  the  observation  of  a  magnitude  very  much  greater 
than  itself,  or  is,  as  it  were,  observed  on  a  greatly  enlarged  scale.  In 
consequence,  a  very  minute  error  is  entailed  on  R  Z  by  the  commission 
of  even  a  considerable  ono  in  Q  R  S :  3dly,  that  in  this  mode  of  obser- 
vation all  the  merely  instrumental  errors  which  affect  the  ordinary  use  of 
the  transit  instrument  are  either  uninfluential  or  eliminated  by  simply 
reversing  the  axis. 

(249.)  To  determine  the  latitude  of  a  station,  then,  is  easy.  It  is 
otherwise  with  its  longitude,  whose  exact  determination  is  a  matter  of  moi'e 
difficulty.  The  reason  is  this :  —  as  there  are  no  meridians  marked  upon 
the  earth,  any  more  than  parallels  of  latitude,  we  are  obliged  in  this  case, 
as  in  the  case  of  the  latitude,  to  resort  to  marks  external  to  the  earth,  i.  e. 
to  the  heavenly  bodies,  for  the  objects  of  our  measurement ;  but  with  this 
difference  in  the  two  cases  —  to  observers  situated  at  stations  on  the  same 
meridian  (i.  e.  differing  in  latitude)  the  heavens  present  different  aspects 
at  all  moments.  The  portions  of  them  which  become  visible  in  a  com- 
plete diurnal  rotation  are  not  the  same,  and  stars  which  are  common  to 
both  describe  circles  differently  inclined  to  their  horizons,  and  differently 
divided  by  them,  and  attain  different  altitudes.  On  the  other  hand,  to 
observers  situated  on  the  same  parallel  (i.  e.  differing  only  in  longitude) 
the  heavens  present  the  same  aspects.  Their  visible  portions  are  tho 
same ;  and  the  same  stars  describe  circles  equally  inclined,  and  similarly 
divided  by  their  horizons,  and  attain  ths  same  altitudes.    In  the  former 


y 


142 


OUTLINES   OF   ASTRONOMY. 


caso  tbcro  is,  in  the  latter  there  ia  not,  any  thing  in  the  appearance  of  the 
heavens,  uratchcd  through  a  whole  diurnal  rotation,  which  indicates  a  dif- 
foronco  of  locality  in  the  observer. 

(250.)  But  not  two  observers,  at  different  points  of  the  earth's  surface, 
can  have  at  the  same  instant  the  same  celestial  hemisphere  visible.  Sup- 
pose, to  fix  our  ideas,  an  observer  stationed  at  a  given  point  of  the  equator, 
and  that  at  the  moment  when  he  noticed  some  bright  star  to  bo  in  his 
zenith,  and  therefore  on  his  meridian,  ho  should  be  suddenly  transported, 
in  an  instant  of  time,  round  one  quarter  of  the  globe  in  a  westerly  direction, 
it  is  evident  that  he  will  no  longer  have  the  same  star  vertically  above 
him :  it  will  now  appear  to  him  to  be  just  rising,  and  he  will  have  to  wait 
six  hours  before  it  again  comes  to  his  zenith,  i.  e.  before  the  earth's  rota- 
tion from  west  to  east  carries  him  buck  again  to  the  line  joining  the  star 
and  the  earth's  centre  from  which  he  set  out. 

(251.)  The  difference  of  the  cases,  then,  may  be  thus  stated,  so  as  to 
afford  a  key  to  the  astronomical  solution  of  the  probUm  of  the  longitude. 
In  the  case  of  stations  differing  only  in  latitude,  the  same  star  comes  to 
the  meridian  at  the  same  time,  but  at  different  altitudes.  In  that  of 
stations  differing  only  in  longitude,  it  comes  to  the  meridian  at  the  same 
altitude,  but  at  different  times.  Supposing,  then,  that  an  observer  is  in 
posscssior.  of  any  means  by  which  he  can  certainly  ascertain  the  time  of  a 
known  star's  transit  across  his  meridian,  he  knows  his  longitude ;  or  if  he 
knows  the  difference  between  its  time  of  transit  across  his  meridian  and 
across  that  of  any  other  station,  he  knows  their  difference  of  longitudes. 
For  instance,  if  the  same  star  pass  the  meridian  of  a  place  A  at  a  certain 
moment,  and  that  of  B  exactly  one  hour  of  sidereal  time,  or  one  twenty- 
fourth  part  of  the  earth's  diurnal  period,  later,  then  the  difference  of  lon- 
gitude between  A  and  B  is  one  hour  of  time  or  15"  of  arc,  and  B  is  so 
much  west  of  A. 

(252.)  In  order  to  a  perfecth  clear  understanding  of  the  principle  on 
which  the  problem  of  finding  the  lon^tude  by  astronomical  observations 
is  resolved,  the  reader  must  kan*  to  distiuguish  between  time,  in  the 
abstract,  as  common  to  the  whofe  nnivtrse.  and  therefore  reckoned  from 
an  epoch  independent  of  local  situation,  and  heal  time,  which  reckons,  at 
each  particular  place,  from  an  epoch,  or  initial  instant,  determined  by  local 
convenience.  Of  time  reckoned  in  the  forraor,  or  abstract  manner,  we 
have  an  example  in  what  wo  have  bn»fore  d?^-  .^  as  equinoctial  time,  which 
dates  from  an  epoch  determined  fey  the  sun  "s  motion  among  the  stars. 
Of  the  latter,  or  local  rtci^oning,  we  have  instances  in  every  sidereal  clock  in 
an  observatory,  and  in  -very  town  clock  for  common  use.  Every  astrono- 
mer regulates,  or  tttmt  u  regulating,  Lis  sidereal  clock,  so  that  it  shall 


DETERMINATION   OF   lONQITUDES. 


143 


indicate  0"  0""  0',when  a  ccrtuiu  poiut  in  tho  heavens,  called  the  equinox, 
is  on  tho  lucridiau  of  hia  station.  This  in  tho  ejwch  of  his  bideroal  time ; 
which  is,  therefore,  entirely  a  local  reckoning.  It  gives  no  information  to 
Bay  that  an  event  happened  at  such  and  such  an  hour  of  sidereal  time, 
unless  wo  particularize  the  station  to  which  the  sidereal  timo  meant  apper- 
tains. Just  80  it  is  with  mean  or  common  timo.  This  is  also  a  local 
reckoning,  having  for  its  epoch  mean  noon,  or  the  average  of  all  the 
times  throughout  tho  year,  when  tho  sun  is  on  the  meridian  o/  that  par- 
ticular place  to  which  it  belortf/s :  and,  therefore,  in  liko  manner,  \>hen 
we  date  any  event  by  mean  time,  it  is  necessary  to  name  the  place,  or 
particularize  what  mean  time  wo  intend.  On  the  other  hand,  a  date 
by  equinoctial  time  is  absolute,  and  requires  no  such  explanatory  addition. 

(253.)  The  astronomer  sets  and  regulates  his  sidereal  clock  by  observ- 
ing the  meridian  passages  of  the  more  conspicuous  and  well-known  stars. 
Each  of  these  holds  in  tho  heavens  a  certain  determinate  and  known  place 
with  respeot  to  that  imaginary  point  called  the  equinox,  and  by  noting  tho 
times  of  their  passage  in  succession  by  his  clock  he  knows  when  the  equi- 
nox passed.  At  that  moment  his  clock  ought  to  have  marked  0''  O"  0'; 
and  if  it  did  not,  he  knows  and  can  correct  its  error,  and  by  the  agreement 
or  disagreement  of  the  errors  assigned  by  each  star  he  can  ascertain 
whether  his  clock  is  correctly  regulated  to  go  twenty-four  hours  in  one 
diurnal  period,  and  if  not,  can  ascertain  and  allow  for  its  rate.  Thus, 
although  his  clock  may  not,  and  imV  nl  cinnot,  either  be  set  correctly,  or 
go  truly,  yet  by  applying  its  •>/•  and  rate  (as  they  are  technically 
termed),  he  can  correct  it*<  luS'  ntions,  and  ascertain  tho  exact  sidereal 
times  corresponding  to  thoik,  aimi  proper  to  his  locality.  This  indispensa- 
ble operation  is  called  gaUii^  Ibis  local  time.  For  simplicity  of  explana- 
tion, however,  we  shall  sappose  the  clock  a  perfect  instrument ;  or,  which 
comes  to  the  same  thipig,  its  error  and  rate  applied  at  every  moment  it  is 
consulted,  and  included  in  its  indications. 

(254.)  Suppose,  ^ow,  of  two  observers,  at  distant  stations,  A  and  B, 
each,  independently  of  the  other,  to  set  and  regulate  his  clock  to  the  true 
sidereal  tane  of  his  station.  It  is  evident  that  if  one  of  these  clocks 
could  be  taken  up  without  deranging  its  going,  and  set  down  by  the  side 
of  the  other,  they  would  be  found,  on  comparison,  to  differ  by  the  exact 
diiTer'^nce  of  their  local  epochs ;  that  is,  by  the  time  occupied  by  the  equi- 
nox, or  by  any  star,  in  passing  from  the  meridian  of  A  to  that  of  B  ;  in 
other  words,  by  their  difference  of  longitude,  expressed  in  sidereal  hours, 
minutes,  and  seconds. 

(255.)  A  pendulum  clock  cannot  be  thus  taken  up  and  transported 
from  place  to  place  without  derangement,  but  a  chronometer  may.     Sup 


'■'.? 


144 


OUTLINES   OF  ASTRONOMY. 


pose,  then,  the  observer  at  B  to  use  a  chronometer  instead  of  a  clock,  he 
may,  by  bodily  transfer  of  the  instrument  to  the  other  station,  procure  a 
direct  comparison  of  sidereal  times,  and  thus  obtain  his  longitude  from  A. 
And  even  if  he  employ  a  clock,  yet  by  comparing  it  first  with  a  good 
chronometer,  and  then  transferring  the  latter  instrument  for  comparison 
with  the  other  clock,  the  same  end  will  be  accomplished,  provided  the 
going  of  the  chronometer  can  be  depended  on. 

(256.)  Were  chronometers  perfect,  nothing  more  complete  and  conve- 
nient than  this  mode  of  ascertaining  differences  of  longitude  could  be 
desired.  An  observer,  provided  with  such  an  instrument,  and  with  a  por- 
table transit,  or  some  equivalent  method  of  determining  ihe  local  time  at 
any  given  station,  might,  by  journeying  from  place  to  place,  and  observing 
the  meridian  passages  of  stars  at  each,  (taking  care  not  to  alter  his  chro- 
nometer, or  let  it  run  down,)  ascertain  their  differences  of  longitude  with 
any  reqtiifed  precision.  In  this  case,  the  sarae  time-keeper  being  used  at 
every  station,  if,  at  one  of  them.  A,  it  mark  true  sidereal  time,  at  any 
other,  B,  it  will  be  just  so  much  sidereal  time  in  error  as  the  difference  of 
longitudes  of  A  and  B  is  equivalent  to :  in  other  words,  the  longitude  of 
B  from  A  will  appear  as  the  error  of  the  time-keeper  on  the  local  time  of 
B.  If  he  travel  westward,  then  his  chronometer  will  appear  continually 
to  gain,  although  it  really  goes  correctly.  Suppose,  for  instance,  he  set 
out  from  A,  when  the  equinox  was  on  the  meiidian,  or  his  chronometer  at 
O',  and  in  twenty-four  hours  (sid.  time)  had  travelled  15°  westward  to  B. 
At  the  moment  of  arrival  there,  his  chronometer  will  again  point  to  O"* ; 
but  the  equinox  will  be,  not  on  his  new  meridian,  but  on  that  of  A,  and 
he  must  wait  one  hour  more  for  its  arrival  at  that  of  B.  When  it  docs  ar- 
rive there,  then  his  watch  will  point  not  to  0'>  but  to  1",  and  will  therefore 
be  1^/ast  on  the  local  time  of  B.  If  he  travel  eastward,  the  reverse  will 
happen. 

(257.)  Suppose  an  observer  now  to  set  out  from  any  station  as  above 
described,  and  constantly  travelling  westward  to  make  a  tour  of  the  globe, 
and  return  to  the  point  he  set  out  from.  A  singular  consequence  will 
happen  :  he  will  have  lost  a  day  in  his  reckoning  of  time.  He  will  enter 
the  day  of  his  arrival  in  his  diary,  as  Monday,  for  instance,  when,  in  fact, 
it  is  Tuesday.  The  reason  is  obvious.  Days  and  nights  are  caused  by  the 
alternate  appearance  of  the  sun  and  stars,  as  the  rotation  of  the  earth  car- 
ries the  spectator  round  to  view  them  in  succession.  So  many  turns  as 
he  makes  absolutely  round  the  centre,  so  often  will  he  pass  through  the 
earth's  shadow,  and  emerge  into  light,  and  so  many  nights  and  days  will 
he  experience.  But  if  he  travel  once  round  the  globe  in  the  direction  of 
its  motion,  he  will,  on  his  arrival,  have  really  made  one  turn  more  round 


DETERMINATION  OF   LONGITUDES. 


145 


its  centre ;  and  if  in  the  opposite  direction,  one  turn  less  than  if  he  had 
remained  upon  one  point  of  its  surface :  in  the  former  case,  then,  he  will 
have  witnessed  one  alternation  of  day  and  night  more,  in  the  latter  one 
less,  than  if  he  had  trusted  to  the  rotation  of  the  earth  alone  to  carry  him 
round,  As  the  earth  revolves  from  west  to  east,  it  follows  that  a  westward 
direction  of  his  journey,  by  which  he  counteracts  its  rotation,  will  cause 
him  to  lose  a  day,  and  an  eastward  direction,  by  which  he  conspires  with 
it,  to  gain  one.  In  the  former  case,  all  his  days  will  be  longer  j  in  the 
latter,  shorter  than  those  of  a  stationary  observer.  This  contingency  has 
actually  happened  to  circumnavigators.  Hence,  also,  it  must  necessarily 
happen  that  distant  settlements,  on  the  same  meridian,  will  differ  a  day 
in  their  usual  reckoning  of  time,  according  as  they  have  been  colonized  by 
settlers  arriving  in  an  eastward  or  in  a  westward  direction, — a  circumstance 
which  may  produce  strange  confusion  when  they  come  to  communicate 
with  each  other.  The  only  mode  of  correcting  the  ambiguity,  and  settling 
the  disputes  which  such  a  difference  may  give  rise  to,  consists  in  having 
recourse  to  the  equinoctial  date,  which  can  never  be  ambiguous. 

(258.)  Unfortunately  for  geography  and  navigation,  the  chronometer, 
though  greatly  and  indeed  wonderfully  improved  by  the  skill  of  modern 
artists,  is  yet  far  too  imperfect  an  instrument  to  be  relied  on  implicitly. 
However  such  an  instrument  may  preserve  its  uniformity  of  rate  for  a 
few  hours,  or  even  days,  yet  in  long  absences  from  home  the  chances  of 
error  and  accident  become  so  multiplied,  as  to  destroy  all  security  of  reli- 
ance on  even  the  best.  To  a  certain  extent  this  may,  indeed,  be  remedied 
by  carrying  out  several,  and  using  them  as  checks  on  each  other  j  but, 
besides  the  expense  and  trouble,  this  is  only  a  palliation  of  the  evil  — 
the  great  and  fundamental, —  as  it  is  the  only  one  to  which  the  determina- 
tion of  longitudes  hy  time-keepers  is  liable.  It  becomes  necessary,  there- 
fore, to  resort  to  other  means  of  communicating  from  one  station  to  another 
a  knowledge  of  its  local  time,  or  of  propagating  from  some  principal  sta- 
tion, as  a  centre,  its  local  time  as  a  universal  standard  with  which  the 
local  time  at  any  other,  however  situated,  may  be  at  once  compared,  and 
thus  the  longitudes  of  all  places  be  referred  to  the  meridian  of  such  cen- 
tral point. 

(259.)  The  simplest  and  most  accurate  method  by  which  this  object 
can  be  accomplished,  when  circumstances  admit  of  its  adoption,  is  that  by 
telegraphic  signal.  Let  A  and  B  be  two  observatories,  or  other  stations, 
provided  with  accurate  means  of  determining  their  respective  local  times, 
and  let  us  first  suppose  them  visible  from  each  other.  Their  clocks  being 
regulated,  and  their  errors  and  rates  ascertained  and  applied,  let  a,  signal 
be  made  at  A,  of  some  sudden  and  definite  kind,  such  as  the  flash  of  gun- 
10 


146 


OUTLINES   OF  ASTRONOMY. 


powder,  the  explosion  of  a  rocket,  the  sudden  extinction  of  a  bright  light, 
or  any  other  which  admits  of  no  mistake,  and  can  be  seen  at  great  dis- 
tances. The  moment  of  the  signal  being  made  must  be  noted  by  each 
observer  at  his  respective  clock  or  watch,  as  if  it  were  the  transit  of  a  star, 
or  other  astronomical  phenomenon,  and  the  error  and  rate  of  the  clock  at 
each  station  being  applied,  the  local  time  of  the  signal  at  each  is  deter- 
luined.  Consequently,  when  the  observers  communicate  their  observations 
of  the  signal  to  each  other,  since  (owing  to  the  almost  instantaneous 
transmission  of  light)  it  must  have  been  seen  at  the  same  absolute  instant 
by  both,  the  difference  of  their  local  times,  and  therefore  of  their  longitudes, 
becomes  known.  For  example,  at  A  the  signal  is  observed  to  happen  at 
5"  0"  0'  sid.  time  at  A,  as  obtained  by  applying  the  error  and  rate  to  the 
time  shown  by  the  clock  at  A,  when  the  signal  was  seen  there.  At  B  the 
same  signal  was  seen  at  5"  4"  0*,  sid.  time  at  B,  similarly  deduced  from 
the  time  noted  by  the  clock  at  B,  by  applying  its  error  and  rate.  Conse- 
quently, the  diflferenco  of  their  local  epochs  is  4'"  0',  which  is  also  their 
difference  of  longitudes  in  time,  or  1°  0'  0"  in  hour  angle. 

(260.)  The  accuracy  of  the  final  determination  may  be  increased  by 
making  and  observing  several  signals  at  stated  intervals,  each  of  which 
affords  a  comparison  of  times,  and  the  mean  of  all  which  is,  of  course, 
more  to  be  depended  on  than  the  result  of  any  single  comparison.  By 
this  means,  the  error  introduced  by  the  comparison  of  clocks  may  be  re- 
garded as  altogether  destroyed. 

(261.)  The  distances  at  which  signals  can  be  rendered  visible  must  of 
course  depend  on  the  nature  of  the  interposed  country.  Over  sea  the 
explosion  of  rockets  may  easily  be  seen  at  fifty  or  sixty  miles ;  and  in 
mountainous  countries  the  flash  of  gunpowder  in  an  open  spoon  may  be 
seen,  if  a  proper  station  be  chosen  for  its  exhibition,  at  much  greater 
distances. 

(262.)  When  the  direct  light  of  the  flash  can  no  longer  be  perceived, 
either  owing  to  the  convexity  of  the  interposed  segment  of  the  earth,  or 
to  intervening  obstacles,  the  sudden  illumination  cast  on  the  under  surface 
of  the  clouds  by  the  explosion  of  considerable  quantities  of  powder  may 
often  be  observed  with  success;  and  in  this  way  signals  have  been  made 
at  very  much  greater  distances.  Whatever  means  can  be  devised  of  exci- 
ting in  two  distant  observers  the  same  sensation,  whether  of  sound,  light, 
or  visible  motion,  at  precisely  the  same  instant  of  time,  may  be  employed 
as  a  longitude  signal.  Wherever,  for  instance,  an  unbroken  line  of  elec- 
tro-telegraphic connection  has  been,  or  hereafter  may  be,  established,  the 
means  exist  of  making  as  complete  a  comparison  of  clocks  or  watches  as 
if  they  stood  side  by  side,  so  that  no  method  more  complete  for  the  deter- 


DETERMINATION  OP  LONGITUDES. 


147 


mination  of  differences  of  longitude  can  be  desired.  The  differences  of 
longiti"'«  between  the  observatories  of  New  York,  Washington,  and  PhUw- 
delpbia,  have  been  very  recently  determined  in  this  manner  by  the  ast*o- 
nomers  at  those  observatories. 

(263.)  Where  no  such  electric  communication  exists,  however,  the 
interval  between  observing  stations  may  be  increased  by  causing  the 
signals  to  be  made  not  at  one  of  them,  but  at  an  intermediate  point ;  for, 
provided  they  are  seen  by  both  parties,  it  is  a  matter  of  indifference  where 
they  are  exhibited.  Still  the  interval  which  could  be  thus  embraced 
would  be  very  limited,  and  the  method  in  consequence  of  little  use,  but 
for  the  following  ingenious  contrivance,  by  which  it  can  be  extended  to 
any  distance,  and  carried  over  any  tract  of  country,  however  difBcult. 

(264.)  This  contrivance  consists  in  establishing,  between  the  extreme 
stations,  whose  difference  of  longitude  is  to  be  ascertained,  and  at  which 
the  local  times  are  observed,  a  chain  of  intermediate  stations,  alternately 
destined  for  signals  and  for  observers.  Thus,  let  A  and  Z  be  the  extreme 
stations.  A  t  « ^.  a  signal  station  be  established,  at  which  rockets,  &c. 
are  fired  at  i.  ■  ^  intervals.  At  C  let  an  observer  be  placed,  provided 
with  a  chronometer ;  at  D,  another  signal  station ;  at  E,  another  observer 
and  chronometer;  till  the  whole  line  is  occupied  by  stations  so  arranged, 
that  the  signal  at  B  can  be  seen  from  A  and  C ;  those  at  D,  from  C  and 
E ;  and  so  on.     Matters  being  thus  arranged,  and  the  errors  and  rates  of 


!> 


Fig.  38. 


A.     B       G      U       E      J? 


the  Ciocks  at  A  and  Z  ascertained  by  astronomical  observation,  let  a  signal 
be  made  at  B,  and  observed  at  A  and  C,  and  the  times  noted.  Thus  the 
difference  between  A's  clock  and  C's  chronometer  becomes  known.  After 
a  short  interval  (five  minutes  for  instance)  let  a  signal  be  made  at  D,  and 
observed  by  C  and  E.  Then  will  the  difference  between  their  respective 
chronometers  be  determined ;  and  the  difference  between  the  former  and 
the  clock  at  A  being  already  ascertained,  the  difference  between  the  clock 
A  and  chronometer  E  is  therefore  known.  This,  however,  supposes  ihat 
the  intermediate  chronometer  C  has  kept  true  sidereal  time,  or  at  least  a 
known  rate,  in  the  interval  between  the  signals.  Now  this  interval  is 
purposely  made  so  very  short,  that  no  instrument  of  any  pretensions  to 


148 


OUTLINES   OF  ASTRONOMY. 


character  can  possibly  produce  an  appreciable  amount  of  error  in  its  lapse 
by  deviations  from  hs  usual  rate.  Thus  the  time  propagated  from  A  to 
C  may  be  considered  as  handed  over,  without  gain  or  loss  (save  from  error 
of  observation),  to  E  Similarly,  by  the  signal  made  at  F,  and  observed 
at  E  and  Z,  the  ti;  so  transmitted  to  E  is  forwarded  on  to  Z ;  and  thus 
at  length  the  clocks  at  A  and  Z  are  compared.  The  process  may  be 
repeated  as  often  as  is  necessary  to  destroy  error  by  a  mean  of  results ; 
and  when  the  line  of  stations  is  numerous,  by  keeping  up  a  succes- 
sion of  signals,  so  as  to  allow  each  observer  to  note  alternately  those  on 
either  side,  which  is  easily  pre-arranged,  many  comparisons  may  be  kept 
running  along  the  line  at  once,  by  which  time  is  saved,  and  other  advan- 
tages obtained.'  In  important  cases  the  process  is  usually  repeated  on 
several  nights  in  succession. 

(265.)  In  pl'ice  of  artificial  signals,  natural  ones,  when  they  occur 
sufficiently  definite  for  observation,  may  be  equally  employed.  In  a  clear 
night  the  number  of  those  singular  meteors,  called  shooting  stars,  which 
may  be  observed,  is  often  very  great,  especially  on  the  9th  and  10th  of 
August,  and  some  other  days,  as  November  12  and  13 ;  and  as  they  are 
sudden  in  their  appearance  and  disappearance,  and  from  the  great  height 
at  which  they  have  been  ascertained  to  take  place  are  visible  over  exten- 
sive regions  of  the  earth's  surface,  there  is  no  doubt  that  they  may  be 
resorted  to  with  advantage,  by  previous  concert  and  agreement  between 
distant  observers  to  watch  and  note  them.'  Those  sudden  disturbances 
of  the  magnetic  needle,  to  which  the  name  of  magnetic  shocks  has  been 
giver,  have  been  satisfactorily  ascertained  to  be,  very  often  at  least, 
simultaneous  over  whole  continents,  and  in  some,  perhaps,  over  the  whole 
globe.  These,  if  observed  at  magnetic  observatories  with  precise  atten- 
tion to  astronomical  time,  may  become  the  means  of  determining  their 
differences  of  longitude  with  more  precision,  possibly,  than  by  any  other 
method,  if  a  sufficient  number  of  remarkable  shocks  be  observed  to 
ascertain  their  identity,  about  which  the  intervals  of  time  between  their 
occurrence  (exactly  alike*  at  both  stations)  will  leave  no  doubt. 


'  For  a  complete  account  of  this  method,  and  the  mode  of  deducing  the  most  advan- 
tageous result  from  a  combination  of  all  the  observations,  see  a  paper  on  the  difference 
of  longitudes  of  Greenwich  and  Paris,  Phil.  Trans.  1826 ;  by  the  Author  of  this 
volume. 

•This  idea  was  first  suggested  by  the  late  Dr.  Maskelyne,  to  whom,  however,  the 
practically  useful  fact  of  their  periodic  recurrence  was  unknown.  Mr.  Cooper  has  thus 
employed  the  meteors  of  the  lOlh  and  12th  August,  1847,  to  determine  the  difference 
uf  longitudes  of  Markree  and  Mount  Eagle,  in  Ireland.  Those  of  the  same  epoch  have 
also  been  used  in  Germany  for  ascertaining  the  longitudes  of  several  stations,  and  with 
very  satisfactory  results. 


LUNAR  METHOD. 


149 


(266.)  Another  species  of  natural  signal,  visible  at  once  over  a  whole 
terrestrial  hemisphere,  is  aflForded  by  the  eclipses  of  Jupiter's  satellites,  of 
which  we  shall  speak  more  at  large  when  we  nnme  to  treat  of  those  bodies. 
Every  such  eclipse  is  an  event  which  possesses  one  great  advantage  in  its 
applicability  to  the  purpose  in  question,  viz.  that  the  time  of  its  happen- 
ing, at  any  fixed  station,  such  as  Greenwich,  can  be  predicted  from  a  long 
coursp  of  previous  recorded  observation  and  calculation  thereon  founded, 
auG  that  this  prediction  is  sufficiently  precise  and  certain,  to  stand  in  the 
place  or  a  corresponding  observation.  So  that  an  observer  at  any  other 
Stat;  on  wherever,  who  shall  have  observed  one  or  more  of  these  eclipses, 
and  ascertained  his  local  time,  instead  of  waiting  for  a  communication 
with  Greenwich,  to  inform  him  at  what  moment  the  eclipse  took  place 
there,  may  use  the  predicted  Greenwich  time  instead,  and  thence,  at 
on',;e,  and  on  the  spot,  determine  his  longitude.  This  mode  of  ascertain- 
ing longitudes  is,  however,  as  will  hereafter  appear,  not  susceptible  of 
great  exactness,  and  should  only  be  resorted  to  when  others  cannot  bo 
had.  The  nature  of  the  observation  also  is  such  that  it  cannot  be  made 
at  sea' ;  so  that,  however  useful  to  the  geographer,  it  is  of  no  advantage 
to  navigation. 

(267.)  But  such  phenomena  as  these  are  of  only  occasional  occurrence; 
and  in  their  intervals,  and  when  cut  off  from  all  communication  with  any 
fixed  station,  it  is  indispensable  to  possess  some  means  of  determining 
longitudes,  on  which  not  only  the  geographer  may  rely  for  a  iinowledge 
of  the  exact  position  of  important  stations  on  land  in  remote  regions,  but 
on  which  the  navigator  can  securely  stake,  at  every  instant  of  his  adven- 
turous course,  the  lives  of  himself  and  comrades,  the  interests  of  his 
country,  and  the  fortunes  of  his  employei-s.  Such  a  method  is  afforded 
by  Lunar  Observations.  Though  we  have  not  yet  introduced  the 
reader  to  the  phenomena  of  the  moon's  motion,  this  will  not  prevent  us 
from  giving  here  th.^  exposition  of  the  principle  of  the  lunar  method ;  on 
the  contrary,  it  will  be  highly  advantageous  to  do  so,  since  by  this  course 
we  shall  have  to  deal  with  the  naked  principle,  apart  from  all  the  peculiar 
sources  of  difficulty  with  which  the  lunar  theory  is  encumbered,  but 

'  To  accomplish  this  is  still  a  desideratum.  Observing  chairs,  suspended  with  stu 
dious  precaution  for  ensuring  freedom  of  motion,  have  been  resorted  to,  under  the  vain 
hope  of  mitigating  the  effect  of  the  ship's  oscillation.  The  opposite  course  seems  more 
promising,  viz.  to  merely  deaden  the  motion  by  a  somewhat  stiff  suspension  (as  by  a 
coarse  and  rough  cable),  and  by  friction  strings  attached  to  weights  running  through 
loops  (not  pulleys)  fixed  in  the  wood-work  of  the  vessel.  At  least,  such  means  have 
been  found  by  the  author  of  singular  efHcacy  in  increasing  personal  comfort  in  the  su8> 
pension  of  a  cot. 


150 


OUTLINES   OF  ASTRONOMY. 


which  are,  in  fact,  completely  extraneous  to  the  jynnciple  of  its  appUca- 
tioD  to  the  problem  of  the  longitudes,  which  is  quite  elementary. 

(268.)  If  there  were  in  the  heavens  a  clock  furnished  with  a  dial-plate 
and  bands,  which  always  marked  Greenwich  time,  the  longitude  of  any 
station  would  be  at  once  uctermined,  so  soon  as  the  local  time  was  known, 
by  comparing  it  with  s  clock.  Now,  the  offices  of  the  dial-plate  and 
hands  of  a  clock  are  .^'.j :  —  the  former  carries  a  set  of  marks  upon  it, 
whose  position  is  known;  the  latter,  by  passing  over  and  among  these 
marks,  inform  us,  by  the  place  it  holds  with  respect  to  them,  what  it  is 
o'clock,  or  what  time  has  elapsed  since  a  certain  moment  when  it  stood  at 
one  particular  spot. 

(269.)  In  a  clock  the  marks  on  the  dial-plate  are  uniformly  distributed 
all  around  ^he  circumference  of  a  circle,  whose  centre  is  that  on  which  the 
hands  revolve  with  a  uniform  motion.  But  it  is  clear  that  we  should,  with 
equal  certainty,  though  with  much  more  trouble,  tell  what  o'clock  it  were, 
if  the  marks  on  the  dial-plate  were  unequally  distributed, —  if  the  hands 
were  ca;centric,  and  their  motion  not  uniform, —  provided  we  knew,  1st, 
the  exact  intervals  round  the  circle  at  which  the  hour  and  minute  marks 
were  placed ;  which  would  be  the  case  if  we  had  them  all  registered  in  a 
table,  from  the  results  of  previous  careful  measurement :  —  2dly,  if  we 
knew  the  exact  amount  and  direction  of  excentricity  of  the  centre  of  mo- 
tion of  the  hands  j  —  and  3dly,  if  we  were  fully  acquainted  with  all  the 
mechanism  which  put  the  hands  in  motion,  so  as  to  be  able  to  say  at  every 
instant  what  were  their  velocity  of  movement,  and  so  as  to  be  able  to  cal- 
culate, without  fear  of  error,  how  much  time  should  correspond  to  so 
MUCH  angular  movement. 

(270.)  The  visible  surface  of  the  starry  heavens  is  the  dial-plate  of  our 
clock,  the  stars  are  the  fixed  marks  distributed  around  its  circuit,  the  moon 
is  the  moveable  hand,  which,  with  a  motion  that,  superficially  considered, 
seems  uniform,  but  which,  when  carefully  examined,  is  found  to  be  far 
otherwise,  and  which,  regulated  by  mechanical  laws  of  astonishing  com- 
plexity and  intricacy  ia  result,  though  beautifully  simple  in  principle  and 
design,  performs  a  monthly  circuit  among  thera,  passing  visibly  over  and 
hiding,  or,  as  it  is  called,  occulting  some,  and  gliding  beside  and  between 
others ;  and  whose  position  among  them  can,  at  any  moment  when  it  is 
visible,  be  exactly  measured  by  the  help  of  a  sextant,  just  as  we  might 
measure  the  place  of  our  clock-hand  among  the  marks  on  its  dial-plate 
with  a  pair  of  compasses,  and  thence,  from  the  known  and  calculated  laws 
of  its  motion,  deduce  the  time.  That  the  moon  does  so  move  among  the 
atarSf  while  the  latter  hold  constantly,  with  respect  to  each  other,  the  same 


LUNAR  METHOD. 


151 


relative  position,  the  notice  of  a  few  nights,  or  even  hours,  will  satisfy  the 
commencing  student,  and  this  is  all  that  at  present  we  require. 

(271.)  There  is  only  one  circumstance  wanting  to  make  our  analogy 
complete.  Suppose  the  hands  of  our  clock,  instead  of  moving  quite  dose 
to  the  dial-plate,  were  considerahly  elevated  above,  or  distant  in  front  of 
of  it.  Unless,  then,  in  viewing  it,  we  kept  our  eye  just  in  the  line  of 
their  centre,  we  should  not  see  them  exactly  thrown  or  projected  upon 
their  proper  places  on  the  dial.  And  if  we  were  either  unaware  of  this 
cause  of  optical  change  of  place,  this  parallax  —  or  negligent  in  not 
takiug  it  into  account  —  we  might  make  great  mistakes  in  reading  the 
time,  by  referring  the  hand  to  the  wrong  mark,  or  incorrectly  appreciating 
its  distance  from  the  right.  On  the  other  hand,  if  we  took  care  to  note, 
in  every  case  when  we  had  occasion  to  observe  the  time,  the  exact  posi- 
tion of  the  eye,  there  would  be  no  difficulty  in  ascertaining  and  allowing 
for  the  precise  influence  of  this  cause  of  apparent  displacement.  Now, 
this  is  just  what  obtains  with  the  apparent  motion  of  the  moon  among 
the  stars.  The  former  (as  will  appear)  is  comparatively  near  to  the  earth 
—  the  latter  immensely  distant ;  and  in  consequence  of  our  not  occupy- 
ing the  centre  of  the  earth,  but  being  carried  about  on  its  surface,  and 
constantly  changing  place,  there  arises  a  parallax,  which  displaces  the 
moon  apparently  among  the  stars,  and  must  be  allowed  for  before  we  can 
tell  the  true  place  she  would  occupy  if  seen  from  the  centre. 

(272.)  Such  a  clock  as  we  have  described  might,  no  doubt,  be  con- 
sidered a  very  bad  one ;  but  if  it  were  our  onli/  one,  and  if  incalculable 
interests  were  at  stake  on  a  perfect  knowledge  of  time,  we  should  justly 
regard  it  as  most  precious,  and  think  no  pains  ill  bestowed  in  studying 
the  laws  of  its  movements,  or  in  facilitating  the  means  of  reading  it 
correctly.  Such,  in  the  parallel  we  are  drawing,  is  the  lunar  theory, 
whose  object  is  to  reduce  to  regularity,  ♦he  indications  of  this  strangely 
irregular-going  clock,  to  enable  us  to  predict,  long  beforehand,  and  with 
absolute  certainty,  whereabouts  among  the  stars,  at  ever}'  hour,  minute, 
and  second,  iu  every  day  of  every  year,  in  Greenwich  local  time,  the 
moon  wotdd  be  seen  from  the  earth's  centre,  and  tcill  be  seen  from  every 
accessible  point  of  its  surface ;  and  such  is  the  lunar  method  of  longi- 
tudes. The  moon's  apparent  angular  distance  from  all  those  principal 
and  conspicViOus  stars  which  lie  in  its  course,  as  seen  from  the  earth's 
centre,,  are  computed  and  tabulated  with  the  utmost  care  and  precision  in 
almanacks  published  under  national  control.  No  sooner  does  an  observer, 
in  any  part  of  the  globe,  at  sea  or  on  land,  measure  its  actual  distance 
from  any  one  of  those  s-iandard  stars  (whose  places  in  the  heavens  havo 
been  ascertained  for  the  purpose  with  the  most  anxious  solicitude,)  than 


!      - 


1 

lit 


1 1 


il' 


152 


OUTLINES  OP  ASTRONOMY. 


I  ill 


he  has,  in  fact,  performed  that  comparison  of  his  local  time  with  the 
local  times  of  every  observatory  in  the  world,  which  enables  him  to  as- 
certain his  difference  of  longitude  from  one  or  all  of  them. 

(273.)  The  latitudes  and  longitudes  of  any  number  of  points  on  the 
earth's  surface  may  be  ascertained  by  the  methods  above  described ;  and 
by  thus  laying  down  a  sufficient  number  of  principal  points,  and  filling  in 
the  intermediate  spaces  by  local  surveys,  might  maps  of  countries  be 
constructed.  In  practice,  however,  it  is  found  simpler  and  easier  to 
divide  each  particular  nation  into  a  series  of  great  triangles,  the  angles 
of  which  are  stations  conspicuously  visible  from  each  other.  Of  these 
triangles,  the  angles  only  are  measured  by  means  of  the  theodolite,  with 
the  exception  of  07ie  side  only  of  orie  triangle,  which  is  called  a  base, 
and  which  is  measured  with  every  refinement  which  ingenuity  can  devise 
or  expense  command.  This  base  is  of  moderate  extent,  rarely  surpassing 
six  or  seven  miles,  and  purposely  selected  in  a  perfectly  horizontal  plane, 
otherwise  conveniently  adapted  to  the  purposes  of  measurement.  Its 
length  between  its  two  extreme  points  (which  are  dots  on  plates  of  gold 
or  platina  let  into  massive  blocks  of  stone,  and  which  are,  or  at  least 
ought  to  be,  in  all  cases  preserved  with  almost  religious  care,  as  monu- 
mental records  of  the  highest  importance,)  is  then  measured,  with  every 
precaution  to  ensure  precision,'  and  its  position  with  respect  to  the 
meridian,  as  well  as  the  geographical  positions  of  its  extremities,  carefully 
ascertained. 
•  (274.)  The  annexed  figure  represents  such  a  chain  of  triangles.    A  B 

Fig.  88. 


i 


is  the  base,  0,  C,  stations  visible  from  both  its  extremities  (one  of  which, 
O,  tve  will  suppose  to  be  a  national  observatory,  with  which  it  is  a  prin- 
cipal object  that  the  base  should  be  as  closely  and  immediately  connected 
as  possible ;)  and  D,  E,  F,  G,  H,  K,  other  stations,  remarkable  points  in 

*  The  greatest  possible  error  in  the  Irish  base  of  between  seven  and  eight  miles, 
Dear  Londonderry,  is  supposed  not  to  exceed  two  inches. 


CONSTRUCTION  OF  MAPS. 


153 


the  country,  by  whose  connection  its  whole  surface  may  bo  covcruci,  as  it 
were,  with  a  network  of  triangles.  Now,  it  is  evident  that  the  angles  of 
the  triangle  A,  B,  C  being  observed,  and  one  of  its  sides.  A,  B,  mea- 
sured, the  other  two  sides,  A  C,  B  C,  may  bo  calculated  by  the  rules  of 
'  trigonometry ;  and  thus  each  of  the  sides  A  C  and  B  C  becomes  in  its 
turn  a  lase  capable  of  being  employed  as  known  sides  of  other  triangles. 
For  instance,  the  angles  of  the  triangles  A  G  G  and  B  C  F  being  known 
by  observation,  and  their  sides  A  0  and  B  C,  we  can  thence  calculate  the 
lengths  A  G,  C  G,  and  B  F,  C  F.  Again,  C  G  and  C  F  being  known 
and  the  included  angle  G  C  F,  G  F  may  be  calculated,  and  so  on.  Thus 
may  all  the  stations  be  accurately  determined  and  laid  down,  and  as  this 
process  may  be  carried  on  to  any  extent,  a  map  of  the  whole  country 
may  be  thus  constructed,  and  filled  in  to  any  degree  of  detail  we  please. 

(275.)  Now,  on  this  process  there  are  two  important  remarks  to  be 
made.  The  first  is,  that  it  is  necessary  to  be  careful  in  the  selection  of 
stations,  so  as  to  form  triangles  free  from  any  veri/  great  inequality  in 
their  angles.  For  instance,  the  triangle  K  B  F  would  be  a  very  improper 
one  to  determine  the  situation  of  F  from  observations  at  B  and  K,  because 
the  angle  F  being  very  acute,  a  small  error  in  the  angle  K  would  produce 
a  great  one  in  the  place  of  F  upon  tJie  line  B  F.  Such  ill-conditioned 
triangles,  therefore,  must  be  avoided.  But  if  this  be  attended  to,  the 
accuracy  of  the  determination  of  the  calculated  sides  will  not  be  much 
short  of  that  which  would  be  obtained  by  actual  measurement  (were  it 
practicable) ;  and,  therefore,  as  we  recede  from  the  base  on  all  sides  as  a 
centre,  it  will  speedily  become  practicable  to  use  as  bases,  the  sides  of  much 
larger  triangles,  such  as  G  F,  G  H,  H  K,  &c.  j  by  which  means  the  next 
step  of  tbo  operation  will  come  to  be  carried  on  on  a  much  larger  scale, 
and  embrace  far  greater  intervals,  than  it  would  have  been  safe  to  do  (for 
the  above  reason)  in  the  immediate  neighbourhood  of  the  base.  Thus  it 
becomes  easy  to  divide  the  whole  face  of  a  country  into  great  triangles  of 
from  30  to  100  miles  in  their  sides  (according  to  the  nature  of  the  ground), 
which,  being  once  well  determined,  may  be  afterwards,  by  a  second  series 
of  subordinate  operations,  broken  up  into  smaller  ones,  and  these  again 
into  others  of  a  still  minuter  order,  till  the  final  filling  in  is  brought  within 
the  limits  of  personal  survey  and  draftsmanship,  and  till  a  map  is  con- 
structed, with  any  required  degree  of  detail. 

(276.)  The  next  remark  we  have  to  make  is,  that  all  the  triangles  in 
question  are  not,  rigorously  speaking,  ^Zane,  but  spherical — existing  on 
the  surface  of  a  sphere,  or  rather,  to  speak  correctly,  of  an  ellipsoid.  In 
very  small  triangles,  of  six  or  seven  miles  in  the  side,  this  may  be 
neglected,  as  the  difference  is  imperceptible^  but  in  the  larger  ones  it 


Il 


lill 


154 


OUTLINES   OF  ASTRONOMY. 


must  be  taken  into  consideration.  It  is  evident  that,  as  every  objeot  used 
for  pointing  the  telescope  of  a  theodolite  has  some  certain  elevation,  not 
only  above  the  soil,  but  above  the  level  of  the  sea,  tnd  as,  moreover, 
those  elevations  differ  in  every  instance,  a  reduction  to  the  horizon  of  all 
the  measured  angles  would  appear  to  be  required.  But,  in  fact,  by  th« 
construction  of  the  theodolite  (art.  192),  which  is  nothing  mo^-e  than  an 
altitude  and  azimuth  instrument,  this  reduction  is  made  in  the  very  act 


of  reading  off  the  horizontal  angles.  Let  E  be  the  centre  of  the  earth; 
A,  B,  C,  the  places  on  its  spherical  surface,  to  which  three  station?,  A, 
P,  Q,  in  a  country  are  referred  by  radii  E  A,  E  B  P,  E  C  Q.  If  a  theo- 
dolite bo  stationed  at  A,  the  axis  of  its  horizontal  circle  will  point  to  E 
when  truly  adjusted,  and  its  plane  will  be  a  tangent  to  the  sphere  at  A, 
intersecting  the  radii  E  B  P,  E  C  Q,  at  M  and  N,  above  the  spherical 
surface.  The  telescope  of  the  theodolite,  it  is  true,  is  pointed  in  succes- 
sion to  P,  and  Q;  but  the  readings  off  of  its  azimuth  circle  give  —  not 
the  angle  P  A  Q,  between  the  directions  of  the  telescope,  or  between  the 
objects  P,  Q,  as  seen  from  A ;  but  the  azimuthal  angle  MAN,  which  is 
the  measure  of  the  angle  A  of  the  spherical  triangle  B  A  C.  Hence 
arises  this  remarkable  circumstance, — that  the  sum  of  the  three  observed 
angles  of  any  of  the  great  triangles  in  geodesical  operations  is  always 
found  to  be  rather  vnore  than  180°.  Were  the  earth's  surface  z.  plane,  it 
ought  to  be  exactly  180° ;  and  this  excess,  which  is  called  the  spherical 
excess,  is  so  far  from  being  a  proof  of  incorrectness  in  the  work,  that  it  is 
essential  to  its  accuracy,  and  offers  at  the  same  time  another  palpable 
proof  of  the  earth's  sphericity, 

(277.)  The  true  way,  then,  of  conceiving  the  subject  of  a  triognoniet- 
rica^.  (iurvey,  when  the  spherical  form  of  the  earth  is  taken  into  considera< 


CONSTRUCTION   OF   MAPS. 


166 


tioD,  is  to  regard  the  network  of  triaoglos  with  which  the  country  is 
covered,  as  the  bases  of  an  assemblage  of  pyramids  converging  to  the 
centre  of  the  earth.  The  theodolite  gives  us  the  true  measures  of  the 
angles  included  hy  the  ^.anes  of  these  pt/ramids;  and  the  surfuc"  of  an 
imaginary  sphere  on  the  level  of  the  sea  intersects  them  in  an  assemblage 
of  spherical  triangles,  above  whose  angles,  in  the  radii  prolonged,  the  real 
stations  of  observation  are  raised,  by  the  superficial  inequalities  of  moun- 
tain and  valley.  The  operose  calculations  of  spherical  trigonometry  which 
this  consideration  would  seem  to  render  necessary  for  the  reductions  uf  a 
survey,  are  dispensed  with  in  practice  by  a  very  simple  and  easy  rule, 
called  the  rule  for  the  spherical  excess,  which  is  to  be  found  in  most  works 
on  trigonometry.  If  we  would  take  into  account  the  ellipticity  of  the 
earth,  it  may  also  be  done  by  appropriate  processes  of  calculation,  which, 
however,  are  too  abstruse  to  dwell  upon  in  a  work  like  the  present. 

(278.)  Whatever  process  of  calculation  we  adopt,  the  result  will  be  a 
reduction  to  the  level  of  the  sea,  of  all  the  triangles,  and  the  consequent 
determination  of  the  geographical  latitude  and  longitude  of  every  station 
observed.  Thus  we  are  at  length  enabled  to  construct  maps  of  countries ; 
to  lay  down  the  outlines  of  continents  and  islands ;  the  courses  of  rivers ; 
the  places  of  cities,  towns  and  villages ;  the  direction  of  mountain  ridges, 
and  the  places  of  their  principal  summits ;  and  all  those  details  which,  as 
they  belong  to  physical  and  statistical,  rather  than  to  astronomical  geog- 
raphy, we  need  not  here  dilate  on.  A  few  words,  however,  will  be  neces- 
sary respeciing  maps,  which  are  used  as  well  in  astronomy  as  in  geog- 
raphy. 

(279.)  A  map  is  nothing  more  than  a  representation,  upon  a  plane,  of 
some  portion  of  the  surface  of  a  sphere,  on  which  are  traced  the  particu- 
lars intended  to  be  expressed,  whether  they  be  continuous  outlines  or 
points.  Now,  as  a  spherical  surface'  can  by  no  contrivance  be  extended 
or  projected  into  a  plane,  without  undue  enlargement  or  contraction  of 
some  parts  in  proportion  to  others ;  and  as  the  system  adopted  in  so  ex- 
tending or  projecting  it  will  dcv-ide  wJiat  parts  shall  be  enlarged  or  rek- 
tively  contracted,  and  in  what  proportions ;  it  follows,  that  when  large 
portions  of  the  sphere  are  to  be  mapped  down,  a  great  difference  in  their 
representations  may  subsist,  according  to  the  system  of  projection  adopted. 

(280.)  The  projections  chiefly  used  in  maps,  are  the  orthoyrapMc, 
stercographic,  and  Mercator's.  In  the  orthographic  projection,  every 
point  of  the  hemisphere  is  referred  to  its  diametral  plane  or  base,  by  a 


k'-i^ 


I  u 


'  We  here  neglect  the  ellipticity  of  the  earth,  which,  for  such  a  purpose  as  map- 
making,  is  too  trifling  to  have  any  material  influence. 


156 


I 


OUTLINES   OF  ASTRONOMT. 
Fig.  41. 


perpendicular  let  fall  on  it,  so  that  the  representation  of  the  hemisphere 
thus  mapped  on  its  base,  is  such  as  would  actually  appear  to  an  eye  placed 
at  an  infinite  distance  from  it.  It  is  obvious,  from  the  annexed  figure, 
that  in  this  projection  only  the  central  portions  are  represented  of  their 
true  forms,  while  all  the  exterior  is  more  and  more  distorted  and  crowded 
together  as  we  approach  the  edges  of  the  map.  Owing  to  this  cause,  the 
orthographic  projection,  though  very  good  for  small  portions  of  the  globe, 
is  of  little  service  for  large  ones. 

(281.)  The  stercoijraphtc  projection  is  in  great  measure  free  from  this 
dv-f'ct.  To  understand  this  projection,  wo  must  conceive  an  eye  to  be 
placed  at  E,  one  extremity  of  a  diameter,  E  C  B,  of  the  sphere,  and  to 
view  the  concave  surface  of  the  sphere,  every  point  of  which,  as  P,  is 
referred  to  the  diametral  plane  A  D  F,  perpendicular  to  E  B  by  the 
visual  line  P  M  E.     The  stereographio  projection  of  a  sphere,  then,  is  a 


Fig. 

42. 

I 

.  ^    V\M  \  C 

Ir/A 

Pv      ^C^T^ 

Wi^ 

J      ;                               ^**N<>„^          ^ 

ly 

true  perspective  representation  of  its  concavity  on  a  diametral  plane; 
and,  as  such,  it  possesses  some  singulsyly  elegant  geometrical  properties, 
of  which  we  shall  state  one  or  two  of  the  principal. 

(282.)  And  first,  then,  all  circles  on  the  sphere  are  represented  by 


PROJECTIONS   OP  THE   SPHERE. 


167 


circles  in  the  projection.  Thus  the  circle  X  is  projected  into  x.  Only 
great  circles  passing  through  the  vertex  B  are  projected  into  straight  lines 
traversing  the  centre  C :  thus,  B  P  A  is  projected  into  0  A. 

2dly.  Every  very  small  triangle,  Or  H  K,  on  the  sphere,  is  represented 
by  a  similar  triangle,  g  h  k,  in  the  projection.  This  is  a  very  valuable 
property,  as  it  insures  a  general  similarity  of  appearance  in  tho  map  to 
the  reality  in  all  its  parts,  and  enables  us  to  project  at  least  a  hemisphere 
in  a  single  map,  without  any  violent  distortion  of  the  configurations  on 
the  surface  from  their  real  forms.  As  in  the  orthographic  projection,  the 
borders  of  the  hemisphere  are  unduly  crowded  together ;  in  tho  stereo- 
graphic,  their  projected  dimensions  are,  on  the  contrary,  somcwhut  ukrgcd 
in  receding  from  the  centre. 

(283.)  Both  these  projections  may  bo  considered  natural  ones,  inap- 
much  as  they  are  really  perspective  representations  of  the  surfac'  on  a 
plane.  Mercator's  is  entirely  an  artificial  one,  representing  the  sphere  as 
it  cannot  be  seen  from  any  one  point,  but  as  it  might  be  seen  by  an  eye 
larricd  successively  over  every  part  of  it.  In  it,  tho  degrees  of  longiiude, 
and  those  of  latitude,  bear  always  to  each  other  their  due  proportion :  the 

Fig.  48. 


U 


equator  is  conceived  to  be  extended  out  into  a  straight  line,  and  the  meri- 
dians are  straight  lines  at  right  angles  to  it,  as  in  the  figure.  Altogether, 
the  general  character  of  maps  on  this  projection  is  not  very  dissimilar  to 
what  would  be  produced  by  referring  every  point  in  tae  globe  to  a  circum- 
scribing cylinder,  by  lines  drawn  from  the  centre,  and  then  unrolling  the 
cylinder  into  a  plane.  Like  the  stereographio  projection,  it  gives  a  true 
representation,  as  to  form,  of  every  particular  small  part,  but  varies 
greatly  in  point  of  scale  in  its  different  regions;  the  polar  portions  in 
particular  being  extravagantly  enlarged ;  and  the  whole  map,  even  of  a 
single  hemisphere,  not  being  comprisable  within  any  finite  limits. 

(284.)  We  shall  not   of  course,  enter  here  into  any  geographical 
details ;  but  one  result  of  maritime  discovery  on  the  great  scale  is,  so  to 


^'7     U      f<      Jfr:^ 


1.  'fc  sii 


158 


OUTLINES   OF  ASTRONOMY. 


speak,  massive  enough  to  coll  for  mention  as  an  astronomical  feature. 
When  the  continents  and  seas  are  laid  down  on  a  globe  (and  since  the 
discovery  of  Australia  and  the  recent  addition  to  our  antarctic  knowledge 
of  Victoria  Land  by  Sir  J.  C.  Ross,  we  are  sure  that  no  very  extensive 
tracts  of  land  remain  unknown),  we  find  that  it  is  possible  so  to  divide 
the  globe  into  two  hemispheres,  that  one  shall  contain  'marly  all  the  land; 
the  other  being  almost  entirely  sea.  It  is  a  fact,  not  a  little  interesting 
to  Englishmen,  and,  combined  with  our  insular  station  in  that  great  high- 
way of  nations,  the  Atlantic,  not  a  little  explanatory  of  our  commercial 
eminence,  that  London'  occupies  nearly  the  centre  of  the  terrestrial  hemi- 
sphere. Astronomically  speaking,  the  fact  of  this  divisibility  of  the 
globe  into  an  oceanic  and  a  terrestrial  hemisphere  is  important,  as  demon- 
strative of  a  want  of  absolute  equality  in  the  density  of  the  solid  mate- 
rial of  the  two  hemispheres.  Considering  the  whole  mass  of  land  and 
water  as  in  a  state  of  equilibrium,  it  is  evident  that  the  half  which  pro- 
trudes must  of  necessity  be  buoyant;  not,  of  course,  that  we  mean  to 
assert  it  to  be  lighter  than  water,  but,  as  compared  with  the  whole  globe, 
in  a  less  degree  heavier  than  that  fluid.  We  leave  to  geologists  to  draw 
from  these  premises  their  own  cnnclusions  (and  we  think  them  obvious 
enough)  as  to  the  internal  constitution  of  the  globe,  and  the  immediate 
nature  of  the  forces  which  sustain  its  continents  at  their  actual  elevation  j 
but  in  any  future  investigations  which  may  have  for  their  object  to  explain 
the  local  deviations  of  the  intensity  of  gravity,  from  what  the  hypothesis 
of  an  exact  elliptic  figure  would  require,  this,  as  a  general  fact,  ought  not 
to  be  lost  sight  of. 

(285.)  Our  knowledge  of  the  surface  of  our  globe  is  incomplete,  un- 
less it  include  the  heights  above  the  sea  level  of  every  part  of  the  land, 
and  the  depression  of  the  bed  of  the  ocean  below  the  surface  over  all  its 
extent.  The  latter  object  is  attainable  (with  whatever  difficulty,  and 
howsoever  slowly)  by  direct  sounding ;  the  former  by  two  distinct  methods : 
th'^  one  consisting  in  trignometrical  measurement  of  the  differences 
of  level  of  all  the  stations  of  a  survey;  the  other,  by  the  use  of  the 
barometer,  the  principle  of  which  is,  in  fact,  identical  with  that  of  the 
sounding  line.  In  both  cases  we  measure  the  distance  of  the  point  whose 
level  we  would  know  from  the  surface  of  an  equilibrated  ocean :  only  in 
the  one  case  it  is  an  ocean  of  water;  in  the  other,  of  air.     In  the  one 

'  More  exactly,  Falmouth.  The  central  point  of  the  hemisphere  which  contains  the 
maximum  of  land  falls  very  nearly  indeed  upon  this  port.  The  land  in  the  opposite 
hemisphere,  with  exception  of  the  tapering  extremity  of  South  America  and  the 
slender  peninsula  of  Malacca,  is  wholly  insular,  and  were  it  not  for  New  Holland 
«rouId  be  quite  insignificant  in  amount. 


BABOMETBIC  MEASUREMENT   OF  HEIGHTS. 


159 


case  our  sounding  is  real  and  tangible ;  in  the  other,  an  imaginary  one, 
measured  by  the  length  of  the  column  of  quicksilver  the  superincumbent 
air  is  capable  of  counterbalancing. 

(286.)  Suppose  that  instead  of  air,  the  earth  and  ocean  were  covered 
with  oil,  and  that  human  life  could  subsist  under  such  circumstances. 
Let  ABODE  be  a  continent,  of  which  the  portion  ABC  projects 

Fig.  44. 


above  the  water,  but  is  covered  by  the  oil,  which  also  floats  at  an  uniform 
depth  on  the  whole  ocean.  Then  if  we  would  know  the  depth  of  any 
point  D  below  the  sea-level,  we  let  down  a  plummet  from  F.  But,  if 
we  would  know  the  height  of  B  above  the  same  level,  we  have  only  to 
send  up  a  float  from  B  to  the  surface  of  the  oil ;  and  having  done  the 
same  at  C,  a  point  at  the  sea  level,  the  difference  of  the  txoo  float  lines 
gives  the  height  in  question. 

(287.)  Now,  though  the  atmosphere  differs  from  oil  in  not  having  a 
positive  surface  equally  definite,  and  in  not  being  capable  of  carrying  up 
any  float  adequate  to  such  an  use,  yet  it  possesses  all  the  properties  of  a 
fluid  really  essential  to  the  purpose  in  view,  and  this  in  particular, —  that, 
over  the  whole  surface  of  the  globe,  its  strata  of  equal  density  supposed 
in  A  state  of  equilibrium,  are  parallel  to  the  surface  of  equilibrium,  or  to 
what  would  he  the  surface  of  the  sea,  if  prolonged  under  the  continents, 
and  therefore  each  or  any  of  them  has  all  the  characters  of  a  definite 
surface  to  measure  from,  provided  it  can  be  ascertained  and  identified. 
Now,  the  height  at  which,  at  any  station  B,  the  mercury  in  a  barometer 
is  supported,  informs  us  at  once  how  much  of  the  atmosphere  is  incum- 
bent on  B,  or,  in  other  words,  in  what  stratum  of  the  general  atmosphere 
(indicated  by  its  density)  B  is  situated :  whence  we  are  enabled  finally  to 
conclude,  by  mechanical  reasoning,'  at  wiiat  height  above  the  sea-levei 
that  degree  of  density  is  to  be  found  over  the  whole  surface  of  the  globe. 
Such  is  the  principle  of  the  application  of  the  barometer  to  the  measure- 
ment of  heights.     For  details,  the  reader  is  referred  to  other  works.* 

'  Newton's  Princip.  ii.  Prop.  22. 

*  Biot,  Astronomie  Physique,  vol.  iii.  For  tables,  see  the  work  of  Biot  cited.  Also 
those  of  Oltmann,  annually  published  by  the  French  board  of  longitudes  in  their 
Annuaire ;  and  Mr.  Baily's  collection  of  Astronomical  Tables  and  Formuls. 


fll 


H 


-'.m 


I  f 


f'M 


160 


OUTLINES  OF  ASTRONOMY. 


(288.)  We  will  content  ourselves  here  with  a  general  caution  against 
an  implicit  dependence  on  barometric  measurements,  except  as  a  differ- 
ential process,  at  stations  not  too  remote  from  each  other.  They  rely  in 
their  application  on  the  assumption  of  a  state  of  equilibrium  in  the  atmo- 
spheric strata  over  the  whole  globe  —  which  is  very  far  from  being  their 
actual  state  (art.  37.)  Winds,  especially  steady  and  general  currents 
sweeping  over  extensive  continents,  undoubtedly  tend  to  produce  some 
degree  of  conformity  in  the  curvature  of  these  strata  to  the  general  form 
of  the  land-surface,  and  therefore  to  give  an  undue  elevation  to  the  mer- 
curial column  at  some  points.  On  the  other  hand,  the  existence  of 
localities  on  the  earth's  surface  where  a  permanent  depression  of  the 
barometer  prevails  to  the  astonishing  extent  of  nearly  an  inch,  has  been 
clearly  proved  by  the  observations  of  Ermann  in  Siberia  and  of  Koss  in 
the  Antarctic  Seas,  and  is  probably  a  result  of  the  same  cause,  and  may 
be  conceived  as  complementary  to  an  undue  habitual  elevation  in  other 
regions. 

(289.)  Possessed  of  a  knowledge  of  the  height  of  stations  above  the 
sea,  we  may  connect  all  stations  at  the  same  altitude  by  level  lines,  the 
lowest  of  which  will  be  the  outline  of  the  sea-coast;  and  the  rest  will 
mark  out  the  successive  coast-lines  which  would  take  place  were  the  sea 
to  rise  by  regular  and  equal  accessions  of  level  over  the  whole  world,  till 
the  highest  mountains  were  submerged.  The  bottoms  of  valleys,  and  the 
ridge-lines  of  hills  are  determined  by  their  property  of  intersecting  all 
these  level  lines  at  right  angles,  and  being,  subject  to  that  condition,  the 
shortest  and  longest,  that  is  to  say,  the  steepest,  and  the  most  gently 
sloping  courses  respectively  which  can  be  pursued  from  the  summit  tci 
the  sea.  The  former  constitute  the  "water  courses"  of  a  country;  the 
latter  its  lines  of  "water-shed"  by  which  it  is  divided  into  distinct  basins 
of  drainage.  Thus  originate  natural  districts  of  the  most  ineffaceable 
character,  on  which  the  distribution,  limits,  and  peculiarities  of  human 
communities  are  in  a  grc  t  measure  dependent.  The  mean  height  of  the 
continent  of  Europe,  or  that  height  which  its  surface  would  have  were  all 
inequalities  levelled  and  the  mountains  spread  equally  over  the  plains,  is 
uncording  to  Humboldt  671  English  feet;  that  of  Asia,  1137;  of  North 
America,  748;  and  of  South  America,  1151. 


Ifli 


CONSTRUCTION   OF   CELESTIAL  MAPS. 


161 


CHAPTEE  V. 
OF    URANOGRAPHT. 

CONSTRUCTION  OP  CELESTIAL  MAPS  AND  GLOBES  BY  OBSERVATIONS 
OF  RIGHT  ASCENSION  AND  DECLINATION.  —  CELESTIAL  OBJECTS  DIS- 
TINGUISHED INTO  FIXED  AND  ERRATIC. —  OP  THE  CONSTELLATIONS. 
—  NATURAL  REGIONS  IN  THE  HEAVENS. — THE  MILKY  WAY.  —  THE 
ZODIAC.  —  OP  THE  ECLIPTIC.  —  CELESTIAL  LATITUDES  AND  LONGI- 
TUDES. —  PRECESSION  OP  THE  EQUINOXES.  —  NUTATION.  —  ABERRA- 
TION.— REFRACTION.— PARALLAX. — SUMMARY  VIEW  OP  THE  URANO- 
GRAPHICAL  CORRECTIONS. 


(290.)  The  determination  of  the  relative  situations  of  objects  in  the 
heavens,  and  the  construction  of  maps  and  globes  which  shall  truly  re- 
present their  mutual  configurations  as  well  as  of  catalogues  which  shall 
preserve  a  more  precise  numerical  record  of  the  position  of  each,  is  a  task 
at  once  simpler  and  less  laborious  than  that  by  which  the  surface  of  the 
earth  is  mapped  and  measured.  Every  star  in  the  great  constellation 
which  appears  to  revolve  above  us,  constituten,  so  to  speak,  a  celestial  sta- 
tion ;  and  among  these  stations  we  may,  as  upon  the  earth,  triangulate,  by 
measuring  with  proper  instrument  their  angular  distances  from  each 
other,  which,  cleared  of  the  effect  of  refraction,  are  then  in  a  state  for 
laying  down  on  charts,  as  we  would  the  towns  and  villages  of  a  country : 
and  this  without  moving  from  our  place,  at  least  for  all  the  stars  which 
rise  above  our  horizon. 

(291.)  Great  exactness  might,  no  doubt,  be  attained  by  this  means, 
and  excellent  celestial  charts  constructed ;  but  there  is  a  far  simpler  and 
easier,  and  at  tic  same  time,  infinitely  more  accurate  course  laid  open  to 
us  if  we  take  advantage  of  the  earth's  rotation  on  its  axis,  and  by  observ- 
ing each  celestial  object  as  it  passes  our  meridian,  refer  it  separately  and 
independently  to  the  celestial  equator,  and  thus  ascertain  its  place  on  the 
surface  of  an  imaginary  sphere,  which  may  be  conceived  to  revolve  with 
it,  and  on  which  it  may  be  considered  as  projected. 

(292.)  The  right  ascension  and  declination  of  a  point  in  the  heavens 
11 


h 


162 


OUTLINES   OF  ASTRONOMY. 


correspond  to  the  longitude  and  latitude  of  a  station  on  the  earth ;  and 
the  place  of  a  star  on  the  celestial  sphere  is  determined^  when  the  former 
elements  are  known,  just  as  that  of  a  town  on  a  map,  by  knowing  the 
latter.  The  great  advantages  which  the  method  of  meridian  observation 
possesses  over  that  of  triangulatior  from  star  to  star,  are,  then,  1st,  That 
in  it  every  star  is  observed  in  thrt  point  of  its  diurnal  course,  when  it  is 
best  seen  and  least  displaced  b^  rcfrvction.  2dly,  That  the  instruments 
required  (the  transit  and  meridiLa  circle)  are  the  simplest  and  least  liable 
to  error  or  derangement  of  any  used  by  astronomers.  3dly,  That  all  the 
observations  can  be  made  systematically,  in  regular  succession,  and  with 
equal  advantages;  there  being  here  no  question  about  advantageous  or 
disadvantageous  triangles,  &c.  And,  lastly.  That,  by  adopting  this 
course,  the  very  quantities  which  we  should  otherwise  have  to  calculate 
by  long  and  tedious  operations  of  spherical  trigonometry,  and  which  are 
essential  to  the  formation  of  a  catalogue,  are  made  the  objects  of  imme- 
diate measurement.  It  is  almost  needless  to  state,  then,  that  this  is  the 
course  adopted  by  astronomers. 

(293.)  To  determine  the  right  ascension  of  a  celestial  object,  all  that 
is  necessary  is  to  observe  the  moment  of  its  meridian  passage  with  a 
transit  instrument,  by  a  clock  regulated  to  exact  sidertjal  time,  or  reduced 
to  such  by  applying  its  known  error  and  rate.  The  rate  may  be  obtained 
by  repeated  observations  of  the  same  star  at  its  successive  meridian  pas- 
sages. The  error,  however,  requires  a  knowledge  of  the  equinox,  or 
initial  point  from  which  all  right  ascensions  in  the  heavens  reckon,  a<j 
longitudes  do  on  the  earth  from  a  first  meridian. 

(294.)  The  nature  of  this  point  will  be  explained  presently  j  but  for 
the  purposes  of  uranography,  in  so  far  as  they  concern  only  the  actual 
configurations  of  the  stars  inter  se,  a  knowledge  of  the  equinox  is  not  neces- 
sary. The  choice  of  the  equinox,  as  a  zero  point  of  right  ascensions,  is 
purely  artificial,  and  a  matter  of  convenience ;  but  as  on  the  earth,  any 
station  (as  a  national  observatory)  may  be  chosen  for  an  origin  of  longi- 
tides ;  so  in  uranography,  any  conspicuous  star  might  be  selected  as  an 
initial  point  from  which  hour  angles  might  be  reckoned,  and  from  which, 
by  merely  observing  differences  or  intervals  of  time,  the  situation  of  all 
others  might  bo  deduced.  In  practice,  these  intervals  are  afi'ectcd  by 
certain  minute  causes  of  inequality,  which  must  be  allowed  for,  and 
which  will  be  explained  in  their  proper  places. 

(295.)  The  declinations  of  celestial  objects  are  obtained,  1st,  By  ob- 
servation of  their  meridian  altitudes,  with  the  mural  or  meridian  circle, 
or  other  proper  instruments.  This  requires  a  knowledge  of  the  geogra- 
phical latitude  of  the  station  of  observation,  which  itself  is  ooly  to  be 


FIXED  AND   ERRATIC   STARS. 


168 


obtained  by  celestial  observation.  2dly,  And  more  directly,  by  observa- 
tion of  th&ir  polar  distances  on  the  mural  circle,  as  explained  in  art.  170, 
which  is  independent  of  any  previous  determination  of  the  latitude 
of  the  station;  neither,  however,  in  this  case,  does  observation  give 
directly  and  immediately  the  exact  declinations.  The  observations  re- 
quire to  be  corrected,  first  for  refraction,  and  moreover  for  those  minute 
causes  of  inequality  which  have  been  just  alluded  to  in  the  case  of  right 
ascensions. 

(296.)  In  this  manner,  then,  may  the  places,  one  among  the  other,  of 
all  celestial  objects  be  ascertained,  and  maps  and  globes  constructed. 
Now  liCre  arises  a  very  important  question.  How  far  are  these  places 
permanent  ?  Do  these  stars  and  the  greater  luminaries  of  heaven  pre- 
serve for  ever  one  invariable  connection  and  relation  of  place  inter  se,  as 
if  they  formed  part  of  a  solid  though  invisible  firmament;  and,  li'v--  the 
great  natural  land-marks  on  the  earth,  preserve  immutably  tlu;  same 
distances  and  bearings  each  from  the  other?  If  so,  the  most  rational 
idea  we  could  form  of  the  universe  would  be  that  of  an  earth  at  absolute 
rest  in  the  centre,  and  a  hollow  crystalline  sphei'e  circulating  round  it, 
and  carrying  sun,  moon,  and  stars  along  in  its  diurnal  motion.  If  not, 
we  must  dismiss  all  such  notions,  and  inquire  individually  into  the  dis- 
tinct history  of  each  object,  with  a  view  to  discovering  the  laws  of  its 
peculiar  motions,  and  whether  any  and  what  other  connection  subsists 
between  them. 

(297.)  So  far  is  this,  however,  from  being  the  case,  that  observations, 
even  of  the  most  cursory  nature,  are  sufficient  to  show  that  some,  at  least, 
of  the  celestial  bodies,  and  those  the  most  conspicuous,  are  in  a  state  of 
continual  change  of  place  among  the  rest.  In  the  case  of  the  moon, 
indeed,  the  change  is  so  rapid  and  remarkable,  that  its  alteration  of  situa- 
tion with  respect  to  such  bright  stars  as  may  happen  to  be  near  it  may  be 
noticed  any  fine  night  in  a  few  hours ;  and  if  noticed  on  t\ro  successive 
nights,  cannot  fail  to  strike  the  most  careless  observer.  With  the  sun, 
too,  the  change  of  place  among  the  stars  is  constant  and  rapid ;  though, 
from  the  invisibility  of  stars  to  the  naked  eye  in  the  day-time,  it  is  not  so 
readily  recognized,  and  requires  either  the  use  of  telescopies  and  angular 
instruments  to  measure  it,  or  a  longer  continuance  of  observation  to  be 
struck  with  it.  Nevertheless,  it  is  only  necessary  to  call  to  mind  its 
greater  meridian  altitude  in  summer  than  in  winter^  and  the  fact  that  the 
stars  which  come  into  view  at  night  (and  which  are  therefore  situated  in 
an  hemisphere  opposite  to  that  occupied  by  the  sun,  and  having  that 
luminary  for  its  centre)  vary  with  the  season  of  the  year,  to  perceive  that 
a  great  change  must  have  taken  place  in  that  inter  7al  in  its  relative  situa- 


i4' 


1  n 


4%    ';%'fci.K<.^'Sf-"' 


iii 


164 


OUTLINES   01    ASTRONOMY. 


tion  with  respect  to  all  the  stars.  Besides  the  sun  and  moon,  too,  there 
are  several  other  bodies,  called  planets,  which,  for  the  most  part,  appear 
to  the  naked  eye  only  rm  the  largest  and  most  brilliant  stars,  and  which 
offer  the  same  phenomenon  of  a  constant  change  of  place  aunotig  tho 
stars;  now  approaching,  and  now  receding  from,  such  of  them  as  we  may 
refer  them  to  as  marks ;  aurJ,  some  in  longer,  some  in  shorter  iieriods, 
making,  like  the  su  i  and  moon,  the  complete  toar  of  the  heavens. 

(298.)  These,  however,  are  exceptions  to  the  general  rule.  The  imu- 
morable  multitude  of  the  stars  which  aro  distributed  over  the  vault  of  the 
heavens  form  a  constellation,  which  preserves,  not  only  to  the  eye  oi"  l!ie 
casual  observer,  but  to  the  nice  examination  of  the  astii)aomer,  a  uni- 
formity '>i'  aspect  which,  when  contrasted  with  the  perpetaal  change  hi 
the  coiifigurations  of  the  sun,  moon,  and  planets,  may  ivell  be  termed 
invariable.  It  is  li.f^  indeed,  that,  by  the  refinement  of  exact  mciisui';- 
ments  prosecuted  from  age  v  i  ago,  s*  nu)  small  changes  of  apparent  place, 
attributable  to  no  illusi'Ht  md  iu  no  terrestrial  cause,  have  been  detected 
in  many  of  them.  Suou  uie  called,  in  astronomy,  the  proper  motions  of 
the  stars.  Eat  thefje  are  so  excessively  slow,  that  their  accumulated 
amount  (even  in  those  stars  for  which  they  are  greatest)  has  bven  insuflB- 
ciont,  in  tlie  whole  duration  of  astronomical  history,  to  produce  any 
obvious  or  material  alteration  in  the  appearance  of  the  starry  heaN^ons. 

(299.)  This  circumstance,  then,  establishes  a  broad  distinction  of  the 
heavenly  bodies  into  two  great  classes ;  —  the  fixed,  among  which  (unless 
in  a  course  of  observations  continued  for  many  years)  no  change  of  mutual 
situation  can  be  detected;  and  the  erratic,  or  wandering  —  (which  is 
implied  in  the  word  planet')  —  including  the  sun,  moon,  and  planets,  as 
well  as  the  singular  class  of  bodies  termed  comets,  in  whose  apparent 
places  among  the  stars,  and  among  each  other,  the  observation  of  a  few 
days,  or  even  hours,  is  sufficient  to  exhibit  an  indisputable  alteration. 

(300.)  Uranography,  then,  as  it  concerns  the  fixed  celestial  bodies  (or, 
as  they  are  usually  called,  the  fixed  stars),  is  reduced  to  a  simple  marking 
down  of  their  relative  places  on  a  globe  or  on  maps ;  to  the  insertion  on 
that  globe,  in  its  due  place  in  the  great  constellation  of  the  stars,  of  the 
pole  of  the  heavens,  or  the  vanishing  point  of  parallels  to  the  earth's 
axis ;  and  of  the  equator  and  place  of  the  equinox :  points  and  circles 
these,  which,  though  artificial,  and  having  reference  entirely  to  our  earth, 
and  therefore  subject  to  all  changes  (if  any)  to  which  the  earth's  axis  may 
be  liable,  are  yet  so  convenient  in  practice,  that  they  have  obtained  an 
admission  (with  some  other  circles  and  lines),  sanctioned  by  usage,  in  all 
globes  and  planispheres.     The  reader,  however,  will  take  care  to  keep 

'  UXavTiTtii,  a  wanderer. 


OP  THE   CONSTELLATIONS. 


165 


them  separate  in  his  mind,  and  to  familiarize  himself  with  the  idea  rather 
of  tico  or  more  celestial  globes,  superposed  and  fitting  on  each  other,  on 
one  of  which  —  a  real  one  —  are  inscribed  the  stars;  on  the  others  those 
imaginary  points,  lines,  and  circles,  which  astronomers  have  devised  for 
their  own  uses,  and  to  aid  their  calculations ;  and  to  accustom  himself  to 
conceive  in  the  latter  or  nrtificial  spheres  a  capability  of  being  shifted  in 
any  manner  upon  the  surface  of  the  other;  so  that,  should  experience 
demonstrate  (as  it  dues)  that  these  artificial  points  and  lines  are  brought, 
])y  a  slow  motion  of  the  earth's  axis,  or  by  other  secular  variations  (as 
they  are  called),  to  coincide,  at  very  distant  intervals  of  times,  with  dif- 
ferent stars,  he  may  not  be  unprepared  for  the  change,  and  may  have  no 
confusion  to  correct  in  his  notions. 

(301.)  Of  course  we  do  not  here  speak  of  those  uncouth  figures  and 
outlines  of  men  and  monsters,  which  are  usually  scribbled  over  celestial 
globes  and  maps,  and  serve,  in  a  rude  and  barbarous  way,  to  enable  us  to 
talk  of  groups  of  stars,  or  districts  in  the  heavens,  by  names  which, 
though  absurd  or  puerile  in  their  origin,  have  obtained  a  currency  from 
which  it  would  be  difficult  to  dislodge  them.  In  so  far  as  they  have  really  (as 
some  have)  any  slight  resemblance  to  the  figures  called  up  in  imagination 
by  a  view  of  the  more  splendid  "constellations,"  they  have  a  certain  con- 
venience ;  but  as  they  are  otherwise  entirely  arbitrary,  and  correspond  to  no 
9m^!<rfl?  subdivisions  or  groupings  of  the  stars,  astronomers  treat  them  lightly, 
or  altogether  disregard  them,'  except  for  briefly  naming  remarkable  stars,  as 
a  Leonis,  <fi  Scorpii,  &c.  &c.,  by  letters  of  the  Greek  alphabet  attached  to 
them.  The  reader  will  find  them  on  any  celestial  charts  or  globes,  and  may 
compare  them  with  the  heavens,  and  there  learn  for  himself  their  position. 

(302.)  There  are  not  wanting,  however,  natural  districts  in  the  heavens, 
which  offer  great  peculiarities  of  character,  and  strike  every  observer: 
such  is  the  milky  way,  that  great  luminous  band,  which  stretches,  every 
evening,  all  across  the  sky,  from  horizon  to  horizon,  and  which,  when 
traced  with  diligence,  and  mapped  down,  is  found  to  form  a  zone  com- 
pletely encircling  the  whole  sphere,  almost  in  a  great  circle,  which  is  neither 
an  hour  circle,  nor  coincident  with  any  other  of  our  astronomical  gram- 
mata.  It  is  divided  in  one  part  of  its  course,  sending  off  a  kind  of 
branch,  which  unites  again  with  the  main  body,  after  remaining  distinct 
for  about  150  degrees,  within  which  it  suffers  an  interruption  in  its  con- 

'  This  disregard  is  neither  supercilious  nor  causeless.  The  constellations  seem  tc 
have  been  almost  purposely  named  and  deUneated  to  cause  as  much  confusion  and 
inconvenience  as  possible.  Innumerable  snakes  twine  through  long  and  contorted 
areas  of  the  heavens,  where  no  memory  can  follow  them  ;  bears,  lions,  and  fishes, 
large  and  small,  northern  and  southern,  confuse  all  nomenclature,  &c.  A  better  sys 
tem  of  constellations  might  have  been  a  material  help  as  an  artificial  memory. 


m 


iC 


166 


OUTLINES   OF  ASTRONOMY. 


DM   I 

I' 


l!i£ 


tinuity.  This  rcninrkublo  belt  has  maintained,  from  the  earliest  ages,  the 
same  relative  situation  among  the  stars;  and,  vihen  examined  through 
powerful  telescope,  is  found  (wonderful  to  relate  !)  to  consist  entirely  of 
stars  scattered  hy  millions^  like  glittering  dust,  on  the  black  ground  of  the 
general  heavens.  It  will  be  described  more  particularly  in  the  subsequent 
portion  of  this  work. 

(303.)  Another  remarkable  region  in  the  heavens  is  the  zodiac,  not 
from  any  thing  peculiar  in  its  own  constitution,  but  from  its  being  tlio 
area  within  which  the  apparent  motions  of  the  sun,  moon,  and  all  tho 
greater  planets  are  confined.  To  trace  the  path  of  any  one  of  these,  it  is 
only  necessary  to  ascertain,  by  continued  observation,  its  places  at  succes- 
sive epochs,  and  entering  these  upon  our  map  or  sphere  in  sufficient  num- 
ber to  form  a  series,  not  too  far  disjoined,  to  connect  them  by  lines  from 
point  to  point,  as  we  mark  out  the  course  of  a  vessel  at  sea  by  mapping 
down  its  place  from  day  to  day.  Now  when  this  is  done,  it  is  found,  first, 
that  the  apparent  path,  or  track,  of  the  sun  on  the  surface  of  the  heavens, 
is  no  other  than  an  exact  great  circle  of  the  sphere  which  is  called  tho 
ecliptic,  and  which  is  inclined  to  the  equinoctial  at  an  angle  of  about  23° 
28',  intersecting  it  at  two  opposite  points,  called  the  equinoctial  points,  or 
equinoxes,  and  which  are  distinguished  from  each  other  by  the  epithets 
vernal  and  autumnal ;  the  vernal  being  that  at  which  the  sun  crosses  the 
equinoctial  from  south  to  north ;  the  autumnal,  when  it  quits  the  northern 
and  entei-s  the  southern  hemisphere.  Secondly,  that  the  moon  and  all 
the  planets  pursue  paths  which,  in  like  manner,  encircle  the  whole 
heavens,  but  are  not,  like  that  of  the  sun,  great  circles  exactly  returning 
into  themselves  and  bisecting  the  sphere,  but  rather  spiral  curves  of  much 
complexity,  and  desci-ibed  with  very  unequal  velocities  in  their  dif  rent 
parts.  They  have  all,  however,  this  in  common,  that  the  general  direc- 
tion of  their  motions  is  the  same  with  that  of  the  sun,  viz.  from  west  to 
cast,  that  is  to  say,  the  contrary  to  that  in  which  both  they  and  the  stars 
appear  to  be  carried  by  the  diurnal  motion  of  the  heavens  j  and,  more- 
over, that  they  never  deviate  far  from  the  ecliptic  on  either  side,  crossing 
and  recrossing  it  at  regular  and  equal  intervals  of  tj.ae,  and  confining 
themselves  within  a  zone,  or  belt  (the  zodiac  already  spoken  of),  extend- 
ing (with  certain  exceptions  among  the  smaller  planets)  not  further  than 
8"  or  9*'  on  either  side  of  the  ecliptic. 

(304.)  It  would  manifestly  be  useless  to  map  down  on  globes  or  charts 
the  apparent  paths  of  any  of  those  bodies  w>>ich  never  retrace  the  same 
oour.<<e,  and  which,  therefore,  demonstrably,  must  occupy  at  some  one  mo- 
ment or  other  of  their  history,  every  point  in  the  area  of  that  zone  of  the 
neavens  within  which  they  are  circumscribed.    The  apparent  complication 


OP  THE  ECLIPTIC  AND   ZODIAC. 


167 


of  their  movements  arise  (that  of  the  moon  excepted)  from  our  viewing 
them  from  a  station  which  is  itself  in  motion,  and  would  disappear,  could 
we  shift  our  point  of  view  and  observe  them  from  the  sun.  On  the  other 
hand  the  apparent  motion  of  the  sun  is  presented  to  us  under  its  least 
involved  form,  and  is  studied,  from  the  Dtation  we  o  cupy,  to  the  greatest 
advantage.  So  that,  independent  of  the  iuiportanoe  of  that  luminary  to 
us  in  other  respects,  it  is  by  the  investigation  of  the  laws  of  its  motions 
in  the  first  instance  that  we  must  rise  to  a  knowledge  of  those  of  all  the 
o*;her  bodies  of  our  system. 

(805.)  Tlio  eciliptic,  which  is  its  apparent  path  among  the  stars,  is  tra- 
versed by  it  in  the  period  called  the  sidereal  year,  which  consists  of 
365*  6"  9-  9-6",  reckoned  In  mean  solar  time  or  866<»  6*  9»  9-6»  reck- 
oned in  sidereal  time.  The  reason  of  this  difference  (and  it  is  this  which 
constitutes  the  origin  of  the  difference  between  solar  and  sidereal  time) 
is,  that  as  the  sun's  apparent  annual  motion  among  the  stars  is  performed 
in  a  contrary  direction  to  the  apparent  diurnal  motion  of  both  sun  and 
stars,  it  comes  to  the  same  thing  as  if  the  diurnal  motion  of  the  sun  were 
so  much  dower  than  that  of  the  stars,  or  as  if  the  sun  lagged  behind 
them  in  its  daily  course.  When  this  has  gone  on  for  ^  whole  year,  the 
sun  will  have  fallen  behind  the  stars  by  a  whole  circumference  of  the 
heavens  —  or,  in  other  words  —  in  a  year  the  sun  will  have  made  fewer 
diurnal  revolutions,  by  one,  than  the  stars.  So  that  the  same  interval  of 
time  which  is  measured  by  366*  6",  &c.  of  sidereal  time,  will  be  called 
365  days,  6  hours,  &c.,  if  reckoned  in  mean  solar  time.  Thus,  then,  is 
the  proportion  between  tho  mean  solar  and  sidereal  time  established, 
which,  reduced  into  a  decimal  fraction,  is  that  of  100273791  to  1.  The 
measurement  of  time  by  these  different  standards  may  be  compared  to 
that  of  space  by  the  standard  feet,  or  ells  of  two  different  nations  j  the 
proportions  of  which,  once  settled  and  borne  in  mind,  can  never  become 
a  source  of  error. 

(306.)  The  position  of  the  ecliptic  among  the  stars  may,  for  our  pre- 
sent purpose,  bo  regarded  as  invariable.  It  is  true  that  this  is  not  strictly 
the  case;  and  on  comparing  together  its  position  at  present  with  that 
which  it  held  at  the  most  distant  epoch  at  which  we  possess  observations, 
we  find  evidences  of  a  small  change,  which  theory  accounts  for,  and  whose 
nature  will  be  hereafter  explained ;  but  that  change  is  so  excessively  slow, 
that  for  a  great  many  successive  years,  or  even  for  whole  centuries,  this 
circle  may  be  regarded,  for  most  ordinary  purposes,  as  holding  the  same 
position  in  the  sidereal  heavens. 

(307.)  The  'poUi  of  the  ecliptic^  like  those  of  any  other  great  circle 
of  the  sphere,  are  opposite  points  on  its  surface,  equidistant  from   the 


? 


■'^^'m 


m 


\  f 


168 


OUTLINES   OP  ASTRONOMY. 


ecliptic  in  every  direction.  They  are  of  course  not  eoincident  with  thobo 
of  the  equinoctial,  but  removed  from  it  by  au  angular  interval  equal  to 
the  inclination  of  the  ecliptic  to  the  equinoctial  (28°  28'),  which  i»  called 
the  obliquity  of  the  edijttic.  In  the  next  iigure,  M  V  p  represent  the 
north  and  south  polos  (by  which  when  used  without  qualification  we  al- 
ways mean  the  poles  of  the  equinoctial)^  and  E  A  Q  Y  the  cquiuoctiel, 

V  S  A  W  the  ecliptic,  and  K  A,  its  poles  -—  tho  spherical  angle  Q  V  8  Is 
the  obliquity  of  tho  ecliptic,  and  is  equal  in  angular  measure  to  P  K  or 
S  Q.     If  we  suppose  the  sun's  apparent  motion  to  be  in  the  direction 

V  S  A  W,  V  will  be  the  vernal  and  A  the  autumnal  equinox.  S  and  W, 
the  two  points  at  which  tho  ecliptic  is  mu^it  distant  from  the  equiuootial, 
are  termed  solstices,  because,  when  arrived  there,  the  sun  ceases  to  recede 
from  the  equator,  and  (in  that  sense,  so  far  as  its  motion  in  declination  is 
concerned)  to  stand  still  in  the  heavens>  S,  the  point  where  the  bud  has 
the  greatest  northern  declination,  is  called  tho  summer,  and  W,  that  where 
it  is  farthest  south,  tho  winter  solstice.  These  epithets  obviously  have 
their  origin  in  the  dependence  of  the  seasons  on  the  sun's  declination, 
which  will  be  explained  in  the  next  chapter.  The  circle  E  K  P  Q  kj), 
which  passes  through  the  poles  of  the  ecliptic  and  equinoctial,  is  called 
the  solstitial  codive ;  and  a  meridian  drawn  through  the  equinoxes,  P  V 

.  p  A,  the  equinoctial  colure. 

(308.)  Since  the  ecliptic  holds  a  determinate  situation  in  tho  starry 
heavers,  it  may  be  employed,  like  the  equinoctial,  to  refer  the  positions 


of  the  stars  to,  by  circles  drawn  through  them  from  its  poles,  and  there- 
fore perpendicular  to  it.  Such  circles  are  termed,  in  astronomy,  circles 
of  latitude — the  distance  of  a  star  from  the  ecliptic,  reckoned  on  the  circle 
of  latitude  passing  through  it,  is  called  the  latitude  of  the  stars — and  the 


•<r 


N0NAGE8IMAL.      ANGLE  C.    dITUATION. 


169 


arc  of  the  ecliptic  intercepted  between  the  vernal  equinox  and  this  circle, 
its  loiKjitude.  In  the  figure,  X  is  a  star,  P  X  R  a  circle  of  declination 
drawn  through  it,  by  which  it  is  referred  to  the  equinoctial,  and  K  X  T  a 
circle  of  latitude  referring  it  to  the  ecliptic  —  then,  as  V  R  is  the  right 
ascension,  and  R  X  the  declination,  of  X,  so  also  is  V  T  it*  longitude,  and 
T  X  its  latitude.  The  use  of  the  terms  longitude  and  latitude,  in  this 
seni<e,  hcoius  to  have  originated  in  considering  the  ecliptic  as  forming  a 
kind  of  natural  equator  to  the  heavens,  as  the  terrestrial  equator  does  to 
the  earth  —  the  former  holding  an  invariable  position  with  respect  to  the 
stars,  as  the  latter  does  with  respect  to  stations  on  the  earth's  surface. 
The  force  of  this  observation  will  presently  become  apparent. 

(309.)  Knowing  the  right  ascension  and  declination  of  an  object,  we 
may  find  its  longitude  and  latitude,  and  vice  versd.  This  is  a  problem 
of  great  use  in  physical  astronomy  —  the  following  is  its  solution  :  —  In 
our  lust  figure,  E  K  P  Q,  the  solstitial  colure  ia  of  course  90°  distant  from 
V,  the  vernal  equinox,  which  is  one  of  its  poles — so  that  V  II  (the  right 
ascension)  being  given,  and  also  Y  E,  the  arc  E  R,  and  its  measure,  the 
spherical  angle  EPR,  or  KPX,  is  known.  In  the  spherical  triangle 
K  P  X,  then,  we  have  given,  1st,  The  side  P  K,  which,  being  the  distance 
of  the  polos  of  the  ecliptic  and  equinoctial,  is  equal  to  the  obliquity  of 
the  ecliptic  j  2d,  The  side  P  X,  the  jaolar  distance,  or  the  complement 
of  the  declination  R  X ;  and,  3d,  the  included  angle  KPX;  and  there- 
fore, by  spherical  trigonometry,  it  is  easy  to  find  the  other  side  K  X,  and 
the  remaining  angles.  Now  K  X  is  the  complement  of  the  required  lati- 
tude X  T,  and  the  angle  P  K  X  being  known,  and  P  K  V  being  a  right 
angle  (because  S  V  is  90°),  the  angle  X  K  V  becomes  known.  Now 
this  is  no  other  than  the  measure  of  the  longitude  V  T  of  the  object. 
The  inverse  problem  is  resolved  by  the  same  triangle,  and  by  a  process 
exactly  similar. 

(310.)  It  is  often  of  use  to  know  the  situation  of  the  ecliptic  in  the  visible 
heavens  at  any  instant ;  that  is  to  say,  the  points  where  it  cuts  the  horizou,  aiid 
the  altitude  of  its  highest  point,  or,  as  it  is  sometimes  called,  the  nonaffi^liuai 
point  of  the  ecliptic,  as  well  as  the  longitude  of  this  point  on  the  ecliptio 
itself  from  the  equinox.  These,  and  all  questions  referable  to  the  same 
data  and  qusesita,  are  resolved  by  the  spherical  triangle  Z  P  E,  formed  by 
the  zenith  Z  (considered  as  the  pole  of  the  horizon),  the  pole  of  the 
equinoctial  P,  and  the  pole  of  the  ecliptic  E.  The  sidereal  time  being 
given,  and  also  the  right  ascension  of  the  pole  of  the  ecliptic  (which  is 
always  the  same,  viz.  18**  0"  0'),  the  hour  angle  ZPE  of  that  point  is 
known.  Then,  in  this  triangle  we  have  given  P  Z,  the  colatitude  j  P  E, 
the  polar  distance  of  the  pole  of  the  ecliptic,  23°  28',  and  the  angle  ZPE 


no 


OUTLINES   OP   ASTRONOMY. 


\    '    , 


from  which  we  may  find,  let,  the  side  Z  E,  which  is  easily  seen  to  be 
equal  to  the  altitude  of  the  nonagesimal  point  sought ;  and  2dly,  the  angle 
P  Z  E,  which  is  the  azimuth  of  the  polo  of  the  ecliptic,  and  which,  there*- 
fore,  being  added  to  and  subtracted  from  90**,  gives  the  azimuth  of  the 
eastern  and  western  intersections  of  the  ecliptic  with  the  horizon.  Lastly, 
the  longitude  of  the  nonagesimal  point  may  bo  had,  by  calculating  in  the 
same  triangle  the  angle  FEZ,  which  is  its  complement. 

(311.)  The  angle  of  situation  of  a  star  is  the  angle  included  between 
circles  of  latitude  and  of  declination  passing  through  it.  To  determine  it 
in  any  proposed  case,  we  must  resolve  the  triangle  P  S  E,  in  which  are 
given  P  S,  P  E,  and  the  angle  S  P  E,  which  is  the  difference  between  the 
star's  right  ascension  and  18  hours ;  from  which  it  is  easy  to  find  the 
angle  P  S  E  required.  This  angle  is  of  use  in  many  inquiries  in  physical 
astronomy.  It  is  called  in  most  books  on  astronomy,  the  angle  of  posi- 
tion, but  this  expression  has  become  otherwise  and  more  conveniently 
appropriated.     (See  Art.  204.) 

(312.)  The  same  course  of  observations  by  which  the  path  of  the  sun 
among  the  fixed  stars  is  tn\oed,  and  the  ecliptic  marked  out  among  them, 
determines,  of  course,  the  place  of  the  equinox  V  (Fig.  art.  808)  upon 
the  starry  sphere,  at  that  (ime — a  point  of  great  importance  in  practical 
astronomy,  as  it  is  the  Oiigii;  or  zero  point  of  right  ascension.  Now, 
when  this  process  is  repeated  at  considerably  distant  intervals  of  time,  a 
very  remarkable  phenomenon  is  observed ;  viz.  that  the  equinox  does  not 
preserve  a  constant  place  among  the  stars,  but  shifts  its  position,  travel- 
ling continually  and  regularly,  although  with  extreme  slowness,  hack- 
vards,  along  the  ecliptic,  in  the  direction  V  "W  from  east  to  west,  or  the 
contrary  to  that  in  which  the  sun  appears  to  move  in  that  circle.  As 
the  ecliptic  and  equinoctial  are  not  very  much  inclined,  this  motion  of  the 


JU- 


PRECESSION  OF  THE  EQUINOXES. 


171 


eqainox  from  east  to  west  along  tho  former,  oouRpires  (speaking  generally) 
with  the  diurnal  motion,  and  carries  it,  with  reference  to  that  motion, 
continually  in  advance  upon  the  stars :  hence  it  has  acquired  tho  name 
of  the  prectw'um  of  the  equinoxea,  because  the  place  of  the  equinox  among 
tho  stars,  at  every  subsequent  moment,  precedes  (with  reference  to  tho 
diurnal  motion)  that  which  it  held  the  moment  before.  The  amount  of 
this  motion  by  which  the  equinox  travels  backward,  or  retrogrades  (as  it 
is  called),  oa  the  ecliptic,  is  0°  0'  6010"  per  annum^  an  extremely 
minute  quantity,  but  which,  by  its  continual  accumulation  from  year  to 
year,  at  last  makes  itself  very  palpable,  and  that  in  a  way  highly  iiicon* 
venient  to  practical  astronomers,  by  destroying,  in  the  lapse  of  a  moderate 
number  of  years,  the  arrangement  of  their  catalogues  of  stars,  and  making 
it  necessary  to  reconstruct  them.  Since  the  formation  of  the  earliest  cutu- 
logue  on  record,  the  place  of  the  equinox  has  retrograded  already  about 
80°.  The  period  in  which  it  performs  a  complete  tour  of  the  ecliptic,  is 
25,868  years.' 

(313.)  The  immediate  uranographical  effect  of  the  precession  of  the 
equinoxes  is  to  produce  a  uniform  increase  of  longitude  in  all  the  heavenly 
bodies,  whether  fixed  or  erratic.  For  the  vernal  equinox  being  tho  initial 
point  of  longitudes,  as  well  as  of  right  ascension,  a  retreat  of  this  point 
on  the  ecliptic  tells  upon  the  longitudes  of  all  alike,  whether  at  rest  or  in 
motion,  and  produces,  so  far  as  its  amount  extends,  the  appearance  of  a 
motion  in  longitude  common  to  all,  as  if  the  whole  heavens  had  a  slow 
rotation  round  the  poles  of  the  ecliptic  in  the  long  period  above  mentioned, 
similar  to  what  they  have  in  twenty-four  hours  round  those  of  the  equi- 
noctial. This  increase  of  longitude,  the  reader  will  of  course  observe  and 
bear  in  mind,  is,  properly  speaking,  neither  a  real  nor  an  apparent  viove- 
ment  of  the  stars.  It  is  a  purely  technical  result,  arising  from  the  gradual 
shifting  of  the  zero  point  from  which  longitudes  are  reckoned.  Had  a 
fixed  star  been  chosen  as  the  origin  of  longitudes,  they  would  have  been 
invariable. 

(314.)  To  form  a  just  idea  of  this  curious  astronomical  phenomenon, 
however,  we  must  abandon,  for  a  time,  the  consideration  of  the  ecliptic, 
as  tending  to  produce  confusion  in  our  ideas ;  for  this  reason,  that  the 
stability  of  the  ecliptic  itself  among  the  stars  is  (as  alreody  hinted,  art. 
306)  only  approximate,  and  that  in  consequence  its  intersection  with  the 
equinoctial  is  liable  to  a  certain  amount  of  change,  arising  from  its  fluc- 
tuation, which  mixes  itself  with  what  is  due  to  tho  principal  uranogra- 
phical cause  of  the  phenomenon.     This  cause  will  become  at  once  appa- 


'Incipiunt  magni  procedere  menses. — Vir&il,  Pollio. 


172 


OUTLINES   OF  ASTRONOMY. 


rent,  if,  instead  of  regarding  the  equinox,  we  fix  our  attention  on  the  pole 
of  the  equinoctial,  or  the  vanishing  point  of  the  earth's  axis. 

(315.)  The  place  of  this  point  among  the  stars  is  easily  determined  at 
any  epoch,  by  the  most  direct  of  all  astronomical  observations,  —  those 
with  the  meridian  or  mural  circle.  By  this  instrument  we  are  enabled  to 
ascertain  at  every  moment  the  exact  distance  of  the  polar  point  from  any 
three  or  more  stars,  and  therefore  to  lay  it  down,  by  triangulating  from 
these  stars,  with  unerring  precision,  on  a  chart  or  globe,  without  the 
least  reference  to  the  position  of  the  ecliptic,  or  to  any  other  circle  not 
naturally  connected  with  it.  Now,  when  this  is  done  with  proper  dili- 
gence and  exactness,  it  results  that,  although  for  short  intervals  of  time, 
such  as  a  few  days,  the  place  of  the  pole  may  be  regarded  as  not  sensibly 
variable,  yet  in  reality  it  is  in  a  state  of  constant,  although  extremely 
slow  motion;  and,  what  is  still  more  remarkable,  this  motion  is  not 
uniform,  but  compounded  of  one  principal  uniform,  or  nearly  uniform, 
part,  and  other  smaller  and  subordinate  periodical  fluctuations :  the 
former  giving  rise  to  the  phenomena  of  precession  ;  the  latter  to  another 
distinct  phenomenon  called  nutation.  These  tw  :>  phenomena,  it  is  true, 
belong,  theoretically  speaking,  to  one  and  the  same  general  head,  and  are 
intimately  connected  together,  forming  part  of  a  great  and  complicated 
chain  of  consequences  flowing  from  the  earth's  rotation  on  its  axis :  but 
it  will  be  conducive  to  clearness  at  present  to  consider  them  separately. 

(316.)  It  is  found,  then,  that  in  virtue  of  the  uniform  part  of  the 
motion  of  the  pole,  it  describes  a  circle  in  the  heavens  around  the  pole  of 
the  ecliptic  as  a  centre,  keeping  constantly  at  the  same  distance  of  23° 
28'  from  it  in  a  direction  from  east  to  west,  and  with  such  a  velocity,  that 
the  annual  angle  described  by  it,  in  this  its  imaginary  orbit,  is  5010"; 
so  that  the  whole  circle  would  be  described  by  it  in  the  above-mentioned 
period  of  25,868  years.  It  is  easy  to  perceive  how  such  a  motion  of  the 
pole  will  give  rise  to  the  retrograde  motion  of  the  equinoxes ;  for  in  the 
figure,  art.  308,  suppose  the  pole  P  in  the  progress  of  its  motion  in  the 
small  circle  P  0  Z  round  K  to  come  to  0,  then,  as  the  situation  of  the 
equinoctial  E  V  Q  is  determined  by  that  of  the  pole,  this,  it  is  evident, 
must  cause  a  displacement  of  the  equinoctial,  which  will  take  a  new 
situation,  E  U  Q,  90°  distant  in  every  part  from  the  new  position  0  of 
the  pole.  The  point  U,  therefore,  in  which  the  displaced  equinoctial  will 
intersect  the  ecliptic,  i.  e.  the  displaced  equinox,  will  lie  on  that  side  of 
V,  its  original  position,  towards  which  the  motion  of  the  pole  is  directed, 
or  to  the  westward. 

(317.)  The  precession  of  the  equinoxes  thus  conceived,  consists,  ther, 
in  a  real  but  very  slow  motion  of  the  pole  of  the  heavens  among  the 


THE   POLE  t'TAR  NOT  ALWAYS  THE   SAME. 


173 


stars,  in  a  small  circle  round  the  pole  of  the  ecliptic.  Now  this  cannot 
happen  without  producing  corresponding  changes  in  the  apparent  diurnal 
motion  of  the  sphere,  and  the  aspect  which  the  heavens  must  present  at 
very  remote  periods  of  history.  The  pole  is  nothing  more  than  the 
vanishing  point  of  the  earth's  axis.  As  this  point,  then,  has  such  a 
motion  as  we  have  described,  it  necessarily  follows  that  the  earth's  axis 
must  have  a  conical  motion,  in  virtue  of  which  it  points,  successively  to 
every  part  of  the  small  circle  in  question.  We  may  form  the  best  idea 
ot  such  a  motion  by  noticing  a  child's  peg-top,  when  it  spins  not  upright, 
or  that  amusing  toy  the  te-to-tum,  which,  when  delicately  executed,  and 
nicely  balanced,  becomes  an  elegant  philosophical  instrument,  and  ex- 
hibits, in  the  most  beautiful  manner,  the  whole  phenomenon.  The 
reader  will  take  care  not  to  confound  the  variation  of  the  position  of  the 
earth's  axis  in  space  with  a  mere  shifting  of  the  imaginary  line  about 
which  it  revolves,  in  its  interior.  The  whole  earth  participates  in  the 
motion,  and  goes  along  with  the  axis  as  if  it  were  really  a  bar  of  iron 
driven  through  it.  That  such  is  the  case  is  proved  by  the  two  great  facts : 
1st,  that  the  latitudes  of  places  on  the  earth,  or  their  geographical  situa- 
tion with  respect  to  the  poles,  have  undergone  no  perceptible  change  from 
the  earliest  ages.  2dly,  that  the  sea  maintains  its  level,  which  could  not 
be  the  case  if  the  motion  of  the  axis  were  not  accompanied  with  a  motion 
of  the  whole  mass  of  the  earth.' 

(318.)  The  visible  effect  of  precession  on  the  aspect  of  the  heavens 
consists  in  the  apparent  approach  of  some  stars  and  constellations  to  the 
pole  and  recess  of  others.  The  bright  star  of  the  Lesser  Bear,  which  we 
call  the  pole  star,  has  not  always  been,  nor  will  always  continue  to  be, 
our  cynosure :  at  the  time  of  the  construction  of  the  earliest  catalogues  it 
was  12°  from  the  pole — it  is  now  only  1°  24',  and  will  approach  yet 
nearer,  to  within  half  a  degree,  after  which  it  will  again  recede,  and 
slowly  give  place  to  others,  which  will  succeed  in  its  companionship  to 
the  pole.  After  a  lapse  of  about  12,000  years,  the  star  a  Lyrae,  the 
brightest  in  the  northern  hemisphere,  will  occupy  the  remarkable  situa- 
tion of  a  pole  star  approaching  within  about  5°  of  the  pole. 

(319.)  At  the  date  of  the  erection  of  the  Great  Pyramid  of  Gizeh, 
which  precedes  by  3970  years  (say  4000)  the  present  epoch,  the  longi- 
tudes of  all  the  stars  were  less  by  55°  45'  than  at  present.     Calculating 


'  Local  changes  of  the  sea  level,  arising  from  purely  geological  causes,  are  easily 
distinguished  from  that  general  and  systematic  alteration  which  a  shifting  of  the  axis 
of  rotation  would  give  rise  to. 


r  f 


. 


174  OUTLINES   OF  ASTRONOMY. 

from  thia  datum*  the  place  of  the  pole  of  the  heavens  among  the  stars^  it 
will  be  found  to  fall  near  a  Draconis ;  its  distance  from  that  star  being  3° 
44'  25".  This  being  the  most  conspicuous  star  in  the  immediate  neigh- 
bourhood was  therefore  the  pole  star  at  that  epoch.  And  the  latitude  of 
Gizeh  being  just  30"  north,  and  consequently  the  altitude  of  the  north 
pole  there  also  30°,  it  follows  that  the  star  in  question  must  have  had  at 
its  lower  culmination,  at  Gizeh,  an  altitude  of  26°  15'  35".  Now  it  is  a 
remarkable  fact,  ascertained  by  the  late  researches  of  Col.  Vyse,  that  of 
the  nine  pyramids  still  existing  at  Gizeh,  six  (including  all  the  largest) 
have  the  narrow  passages  by  which  alone  they  can  be  entered,  (all  which 
open  out  on  the  northern  faces  of  their  respective  pyramids)  inclined  to 
the  horizon  downwar  Is  at  angles  as  follows. 

1st,  or  Pyramid  of  Cheops  26°  41 

2d,  or  Pyramid  of  Cephren 25    55 

3d,  or  Pyramid  of  Mycerinus 26      2 

4th,  27      0 

5th, 27    12 

9th,  28      0 

Mean    -    26    47 

Of  the  two  pyramids  at  Abousseir  also,  which  alone  exist  in  a  state  of 
suflBcient  preservation  to  admit  of  the  inclinations  of  their  entrance  pas- 
,?ages  being  determined,  one  has  the  angle  27°  5',  the  other  26°. 

(320. )  At  the  bottom  of  every  one  of  these  passages  therefore,  the  then 
pole  star  must  have  been  visible  at  its  lower  culmination,  a  circumstance 
which  can  hardly  be  supposed  to  have  been  unintentional,  and  was  doubt- 
less connected  (perhaps  superstitiously)  with  the  astronomical  observation 
of  that  star,  of  whose  proximity  to  the  pole  at  the  epoch  of  the  erection 
of  these  wonderful  structures,  we  are  thus  furnished  with  a  monumental 
record  of  the  most  imperishable  nature. 

(321.)  The  nutation  of  the  earth's  axis  is  a  small  and  slow  subordinate 
gyratory  movement,  by  which,  if  subsisting  alone,  the  pole  would  describe 
among  the  stars,  in  a  period  of  about  nineteen  years,  a  minute  ellipsis, 
having  its  longer  axis  equal  to  18"-5,  and  its  shorter  to  13"-74 ;  the  longer 
being  directed  towards  the  pole  of  the  ecliptic,  and  the  shorter,  of  course, 
at  right  angles  to  it.     The  consequence  of  this  real  motion  of  the  pole  is 

*  On  this  calrtjiation  the  diminution  of  the  obliquity  of  the  eliptic  in  the  4000  years 
elapsed  has  no  influence.  That  diminution  arises  from  a  change  in  the  plane  of  the 
*;arth's  orbit,  and  has  nothing  to  do  with  the  change  in  the  position  oi  its  axis,  as  re- 
terred  to  the  starry  sphcro. 


NUTATION   OF   THE   EARTH'S  AXIS. 


175 


an  apparent  approach  and  recess  of  all  the  stars  in  the  heavens  to  the 
pole  in  the  same  period.  Since,  also,  the  place  of  the  equinox  on 
the  ecliptic  is  determined  by  the  place  of  the  pole  in  the  heavens,  the 
same  cause  will  give  rise  to  a  small  alternate  advance  and  recess  of  the 
equinoctial  points,  by  which,  in  the  same  period,  both  the  longitudes  and 
the  right  ascensions  of  the  stars  will  be  also  alternately  increased  and 
diminished. 

(322.)  Both  these  motions,  however,  although  here  considered  sepa- 
rately, subsist  jointly  j  and  since,  while  in  virtue  of  the  nutation,  the  polo 
is  describing  its  little  ellipse  of  18" -5  in  diameter,  it  is  carried  by  the 
greater  and  regularly  progressive  motion  of  precession  over  so  much  of  its 
circle  round  the  pole  of  the  ecliptic  as  corresponds  to  nineteen  years, — 
that  is  to  say,  over  an  angle  of  nineteen  times  50"  •!  round  the  centre 
(which,  in  a  small  circle  of  23"  28'  in  diameter,  corresponds  to  6'  20", 
as  seen  from  the  centre  of  the  sphere)  :  the  path  which  it  will  pursue  in 
virtue  of  the  two  motions,  subsisting  jointly,  will  be  neither  an  ellipse 
nor  an  exact  circle,  but  a  gently  undulated  ring  like  that  in  the  figure 
(where,  however,  the  undulations  are  much  exaggerated).  (See  Jig, 
to  art.  325.) 

(323.)  These  movements  of  precession  and  nutation  are  common  to  all 
the  celestial  bodies,  both  fixed  and  erratic ;  and  this  circumstance  makes 
it  impossible  to  attribute  them  to  any  other  cause  than  a  real  motion  of  the 
earth's  axis  such  as  we  have  described.  Did  they  only  affect  the  stars, 
they  might,  with  equal  plausibility,  be  urged  to  arise  from  a  real  rotation 
of  the  starry  heavens,  as  a  solid  shell,  round  an  axis  passing  il.itmgh  the 
poles  of  the  ecliptic  in  25,868  years,  and  a  real  ecliptic  gyrati>ja  of  that 
axis  in  nineteen  y*^ars :  but  since  they  also  affect  the  sun,  moor;,  a;id  pla- 
nets, which,  having  motions  independent  of  the  generd  body  of  the  stars, 
cannot  without  extravagance  be  supposed  attached  to  the  '  lestial  c  jncave,' 
this  idea  falls  to  the  ground ;  and  there  only  remains,  then,  a  real  motion 
in  the  earth  by  which  they  can  be  accounted  for.  It  will  be  shown  in  a 
subsequent  chapter  that  they  are  necessary  consequences  of  the  rotation 
of  the  earth,  combined  with  its  elliptical  figure,  and  the  unequal  attrac- 
tion of  the  sun  and  moon  on  its  polar  and  equatorial  regions. 

(324.)  Uranographically  considered,  as  affecting  the  apparent  places  of 
the  stars,  they  are  of  the  utmost  importance  in  practical  astronomy.  When 
we  speak  of  the  right  ascension  and  declination  of  a  celestial  object,  it 


'  This  argument,  cogent  as  it  is,  acquires  additional  and  decisive  force  from  the  law 
of  nutaiion,  which  is  dependent  on  the  position,  for  the  time,  of  the  lunar  orbit.  H 
we  attribute  it  to  a  real  motion  of  the  celestial  sphere,  we  must  then  maintain  that  sphere 
to  be  kepi  in  a  constant  state  of  tremor  by  the  motion  of  the  moon. 


176 


OUTLINES   or  JiSTRONOMT. 


4 


mi 


becomes  necessary  to  state  what  epoch  we  intend,  and  whether  we  mean 
the  mean  right  ascension  —  cleared,  that  is,  of  the  periodical  fluctuation  in 
its  amount,  which  arises  from  nutation,  or  the  apparent  right  ascension, 
which,  being  reckoned  from  the  actual  place  of  the  vernal  equinox,  is 
aflFected  by  the  periodical  advance  and  recess  of  the  equinoctial  point  pro- 
duced by  nutation  —  and  so  of  the  other  elements.  It  is  the  practice  of 
astronomers  to  reduce,  as  it  is  termed,  all  their  observations,  both  of  right 
ascension  and  declination,  to  some  common  and  fionvenient  epoch  — such 
as  the  beginning  of  the  year  for  temporary  purposes,  or  of  the  decade,  or 
the  century  for  more  permanent  uses,  by  subtracting  from  them  the  whole 
eflFect  of  precession  in  the  interval ;  and,  moreover,  to  divest  them  of  the 
influence  of  nutation  by  investigating  and  subducting  the  amount  of 
change,  both  in  right  ascension  and  declination,  due  to  the  displacement 
of  the  pole  from  the  centre  to  the  circumference  of  the  little  ellipse  above 
mentioned.  This  last  process  is  technically  termed  correcting  or  equatiai^ 
the  observation  for  nutation ;  by  which  latter  word  is  always  understood, 
in  astronomy,  the  getting  rid  of  a  periodical  cause  of  fluctuation,  and  pre- 
senting a  result,  not  as  it  was  observed,  but  as  it  would  have  been  observed, 
had  that  cause  of  fluctuation  had  no  existence. 

(325.)  For  these  purposes,  in  the  present  case,  very  convenient 
formula)  have  been  derived,  and  tables  constructed.  They  are,  however, 
of  too  technical  a  character  for  this  work ;  we  shall,  however,  point  out 
the  manner  in  which  the  investigation  is  conducted.  It  has  been  shown 
in  art.  309  by  what  means  the  right  ascension  and  declination  of  an 
object  are  derived  from  its  longitude  and  latitude.  Tkeferring  to  the 
figure  of  that  article,  and  supposing  tbr  triangle  K  P  X  orthographically 
projected  on  the  plane  of  the  ^-liptic  as  in  the  annexed  figure :  in  the 
triangle  KPX,  KP  is  the  Mi.juity  of  the  ecliptic,  KX  the  co-latitude 
(or  complement  of  latitude),  and  the  angle  P  K  X  the  co-lonr/itude  of  the 
object  X.  These  are  the  data  of  our  question,  of  which  the  second  is 
constant,  and  the  other  two  are  varied  by  the  effect  of  precession  and 
nutation :  and  their  variations  (^considering  the  minuteness  of  the  latter 
eflect  generally,  and  the  small  number  of  years  in  c  mparison  of  the 
whole  period  of  25,808,  for  which  we  ever  require  to  estimate  the  effect 
of  the  former,  are  of  that  order  which  may  be  regarded  as  infinitesimal 
in  geometry,  aid  treated  as  such  without  fear  <^  error.  The  whole  ques- 
tion, then,  is  reduced  to  this: — In  a  spherical  triangle  K  PX,  in  which 
one  side  K  X  is  constant,  and  an  angle  K,  and  adjacent  sid*  K  P  vary  by 
given  infinitesimal  changes  of  the  position  of  P :  reqaiw^  the  efosmges 
thence  arising  in  the  other  side  F X,  and  thf  angle  KPX.  Thiir  m  • 
very  simple  and  easy  oroblem  of  tiflkmltid  gfoK^try,  and  mug  k 


CORRECTIONS  FOR   PRECESSION  AND  NUTATION.  177 

Fig.  47. 


it  gives  at  once  the  reductions  we  are  seeking  j  for  P  X  being  the  polar 
distance  of  the  object,  and  the  angle  K  P  X  its  right  ascension  plus  90", 
their  variations  are  the  very  quantities  we  seek.  It  only  remains,  then, 
to  express  in  proper  form  the  amount  of  the  precession  and  nutation  in 
longitude  and  latitude,  when  their  amount  in  right  ascension  and  declina- 
tion will  immediately  be  obtained. 

(326.)  The  precession  in  latitude  is  zero,  since  the  latitudes  of  objects 
are  not  changed  by  it :  that  in  longitude  is  a  quantity  proportional  to  the 
time  at  the  rate  of  50"-10  per  annum.  With  regard  to  the  nutation  in 
longitude  and  latitude,  these  are  no  other  than  the  abscissa  and  ordinate 
of  the  little  ellipse  in  which  the  pole  moves.  The  law  of.  its  motion, 
however,  therein,  cannot  be  understood  till  the  reader  has  been  made 
acquainted  with  the  principal  features  of  the  moon's  motion  on  which  it 
depends. 

(327.)  Another  consequence  of  what  has  been  shown  respecting  pre- 
cession and  nutation  is,  that  sidereal  time,  as  astronomers  use  it,  i.  e.  as 
reckoned  from  the  transit  of  the  equinoctial  point,  is  not  a  mean  or  %ini- 
formly  floioing  quantity,  being  affected  by  nut^ation ;  and  moreover,  that 
so  reckoned,  even  when  cleared  of  the  periodical  fluctuation  of  nutation, 
it  does  not  strictly  correspOiid  to  the  earth's  diurnal  rotation.  As  the  sun 
lo%e%  one  day  in  the  year  on  the  stars,  bv  its  direct  motion  in  longitude ; 
so  the  equinox  gains  one  day  in  25,868  years  on  them  by  its  retrogradc- 
fi'!n.  We  oughtj  therefore,  as  carefully  to  distinguish  between  mean  and 
apparent  sidereal  as  between  mean  and  apparent  solar  time. 

(328.)  Neither  precession  nor  nutation  changes  the  apparent  places  of 
oelential  objects  inter  se.     Yfe  see  them,  so  far  aa  these  causes  go,  as  they 
12 


'    V 


I'  a 


178 


OUTLINES   OF  ASTRONOMY. 


m 


liilL 


are,  though  from  a  station  more  or  less  unstable,  as  we  see  distant  land 
objects  correctly  formed,  though  appearing  to  rise  and  fall  when  viewed 
from  the  heaving  deck  of  a  ship  in  the  act  of  pitching  and  rolling.  But 
there  is  an  optical  cause,  independent  of  re*;dction  or  of  perspective,  which 
displaces  them  one  among  the  other,  and  causes  us  to  view  the  heavens 
under  an  aspect  always  to  a  certain  slight  extent  false ;  and  whose  in- 
fluence must  be  estimated  and  allowed  for  before  we  can  obtain  a  precise 
knowledge  of  the  place  of  any  object.  This  cause  is  what  is  called  the 
aberration  of  light ;  a  singular  and  surprising  effect  arising  from  this,  that 
we  occupy  a  station  not  at  rest  but  in  rapid  motion ;  and  that  the  apparent 
cl! "Cottons  of  the  rays  of  light  are  not  the  same  to  a  spcx.  ator  in  motion 
as  to  one  at  rest.  As  the  estimation  of  its  effect  belongs  to  uranography, 
V.C  :  I  list  explain  it  here,  though,  in  so  doing,  we  must  anticipate  some  of 
tiio  results  to  be  detailed  in  subsequent  chapters. 

(329.)  Suppose  a  shower  of  rain  to  fall  perpendicularly  in  a  dead  calm ; 
a  person  exposed  to  the  shower,  who  would  stand  quite  still  and  upright, 
wt  ukl  ieceive  the  drops  on  his  hat,  which  would  thus  shelter  him,  but  i^ 
he  ran  forward  in  any  direction  they  would  strike  him  in  the  face.  The 
effect  would  be  the  same  as  if  he  remained  still,  and  a  wind  should  arise 
of  the  same  velocity^  and  drift  them  against  him.     Suppose  a  ball  let  fall 


from  a  point  A  above  a  horizontal  line  E  F,  and  that  at  B  were  placod  to 
receive  it  the  open  mouth  of  an  inclined  hollow  tube  P  Q ;  if  the  lube 
were  held  immoveable  the  ball  would  strike  on  its  lower  side,  but  if  the 
tube  were  carried  forward  in  the  directiou  E  F,  with  a  velocity  properly 
adjusted  at  every  instant  to  that  of  the  ball,  YihWa  preserving  its  inclina- 
tion to  the  horizon,  so  that  when  tlie  ball  in  its  natural  descent  reached 
C,  tl  e  tube  should  have  been  earned  into  the  position  R  S,  it  is  evident 
that  'he  ball  would,  throughout  its  whole  descent,  be  found  in  the  axis  of 


ABERRATION   OF  LIGHT. 


179 


the  tube ;  and  a  spectator  referring  to  the  tube  the  motion  of  the  ball, 
and  carried  along  with  the  former,  unconscious  of  its  motion,  would  fancy 
that  the  ball  had  been  moving  in  the  inclined  direction  R  S  of  the  tube's 
axis. 

(330.)  Our  eyes  and  telescopes  are  such  tubes.  In  whatever  manner 
we  consider  light,  whether  as  an  advancing  wave  in  a  motionless  ether,  or 
a  shower  of  atoms  traversing  space,  (provided  that  in  both  cases  we  regard 
it  as  absolutely  incapable  of  suffering  resistance  or  corporeal  obstruction 
from  the  particles  of  transparent  media  traversed  by  it,')  if  in  the  interval 
between  the  rays  traversing  the  object  glass  of  the  one  or  the  cornea  of 
the  other  (at  which  moment  they  acquire  that  convergence  which  directs 
them  to  a  certain  point  in  fixed  space)^  and  their  arrival  at  their  focus, 
the  cross  wires  of  the  one  or  the  retina  of  the  other,  be  slipped  aside,  the 
point  of  convergence  (which  remains  unchanged)  will  no  longer  corres- 
pond to  the  intersection  of  the  wires  or  the  central  point  of  our  visual 
areu.  The  object  then  will  appear  displaced  j  and  the  amount  of  this 
displacement  is  aberration. 

(331.)  The  earth  is  moving  through  space  with  a  velocity  of  about  19 
miles  per  second,  in  an  elliptic  path  round  the  sun,  and  is  therefore 
changing  the  direction  of  its  motion  at  every  instant.  Light  travels  with 
a  velocity  of  192,000  miles  per  second,  which,  although  much  greater  than 
that  of  the  earth,  is  yet  not  infinitely  so.  Time  is  occupied  by  it  in 
traversing  any  space,  and  in  that  time  the  earth  describes  a  space  which  is 
to-  the  former  as  19  to  192^000,  or  as  the  tangent  of  20"-5  to  radius. 
Suppose  now  A  P  S  to  represent  ray  of  light  from  a  star  at  A,  and  let 
the  tube  PQ  be  that  of  a  telescope  so  incliucu  a. -ward  that  the  focus 
formed  by  its  object-glass  shall  be  received  upon  its  cross  wire,  it  i& 
evident  from  what  has  been  said,  that  the  inclination  of  the  tube  must 
be  such  as  to  make  P  S  :  S  Q  : :  velocity  of  light :  velocity  of  the  earth  : : 
1 :  tan.  20'' '5 ;  and,  therefore,  the  angle  S  P  Q,  or  P  S  R,  by  which  the 
axis  of  the  telescope  must  deviate  from  the  true  direction  of  the  star, 
Tiiust  be  20"-5. 

(882.)  A  similar  re:iSoning  will  bold  good  when  the  direction  of  the 
earth's  motion  is  not  ptrpendicuW  >/i  the  visual  raj.     If  3  B  be  the  true 


.':  >  ; 


m 


'  This  condition  is  indispennable.  Without  it  we  fall  into  all  those  difficiiltiea  which 
M.  Do;)(,ier  has  so  well  pointea  out  in  his  paper  on  Aberration  (Abhandlungen  der  k. 
i.oemischen  Gesell  ,i-haft  der  WisBenschafu-n.  Folge  V.  vol.  iii.).  If  light  itself,  or 
■.he  lur.iiiiiferous  e'.her,  be  corporeal,  'he  oondiiion  insisted  on  amounts  to  a  formal  sur- 
render of  the  dogmii,  either  of  the  extension  or  of  the  impenetrability  of  matter ;  Rt 
least  in  the  sense  in  »vhich  those  terais  have  b'^en  hitherto  used  by  metaphysicians. 
At  the  point  to  which  science  is  arrived,  probably  few  will  be  found  disposed  to  main 
tain  either  the  one  or  'iie  other. 


Ml 

Pa"'' 

1 1 


'.««■■' 
/!:! 


180 


OUTLINES  07  ASTRONOMT. 


Fig.  48. 


u 


direction  of  the  visual  ray,  and  A  C  the  position  in  which  the  telobcope 
requires  to  be  held  in  tho  apparent  direction,  we  must  still  have  the  pro- 
portion B  0  :  B  A  :  :  velocity  of  light :  velocity  of  the  earth  : :  rad. : 
sine  of  20" -5  (for  in  such  small  angles  it  matters  not  whether  we  use  the 
sines  or  tangents).  But  we  have,  also,  by  trigonometry,  B  C  :  B  A  : : 
sine  of  BAG:  sine  of  ACB  or  CBD,  which  last  is  i\o  apparent  dis- 
placement caused  by  aberration.  Thus  it  appears  that  he  sine  of  the 
aberration,  or  (since  the  angle  is  extremely  small)  the  aberration  itself,  ia 
proportional  to  the  sine  of  the  angle  made  by  the  earth's  motion  in  space 
with  the  visual  ray,  and  is  therefore  a  maximum  when  the  line  of  sight  is 
perpendicular  to  the  direction  of  the  earth's  motion. 

(333.)  The  uranographical  effect  of  aberration,  then,  is  to  distort  the 
aspect  of  the  heavens,  causing  all  the  stars  to  crowd  as  it  were  dhoctly 
towards  that  point  in  tho  heavens  which  is  the  vanishing  point  of  all  linea 
parallel  to  that  in  which  the  earth  is  for  the  time  moving.  As  the  earth 
moves  round  the  sun  in  the  plane  of  the  ecliptic,  this  point  must  lie  in 
that  plane,  90°  in  advance  of  the  earth's  longitude,  or  90*^  behind  the 
sun's,  and  shifts  of  course  continually,  describing  the  circumference  of  the 
ecliptic  in  a  year.  It  is  easy  to  demonstrate  that  the  effect  on  each  par- 
ticular star  will  be  to  make  it  apparently  describe  a  small  ellipse  in  the 
heavens,  having  for  its  centre  the  point  in  which  the  star  would  be  seen 
if  vhe  earth  were  at  rest. 

(334.)  Aberration  then  affects  the  apparent  right  ascensions  and  decli- 
nations of  all  the  stars,  and  that  by  quantities  easily  calculable.  The 
formulae  most  convenient  for  that  purpose,  and  which,  systematically 
embracing  at  the  same  time  the  corrections  for  precession  and  nutation, 
enable  the  observer,  with  the  utmost  readiness,  to  disencumber  his  obser- 
vations of  right  ascension  and  declination  of  their  influence,  have  been 
constructed  by  Prof.  Bessel,  and  tabulated  in  the  appendix  to  the  first, 
volume  of  the  Transactions  of  the  Astronomical  Society,  where  they  will 
be  found  accompanied  with  an  extensive  catalogue  of  the  places,  for  1830, 


ABERRATION   OF  LIGHT. 


181 


of  the  principal  fixed  stars,  one  of  the  most  useful  and  beat  arranged  worka 
of  the  kind  which  has  ever  appeared. 

(335.)  When  the  body  from  which  the  visual  ray  emanate*  Is  itself  in 
motien,  an  effect  arises  which  is  not  properly  speaking  aberration,  though 
it  is  usually  treated  imder  that  head  in  astronomical  books,  and  indeed 
confounded  with  it,  to  the  production  of  some  confusion  in  the  mind  of 
the  student.  The  eflFect  in  question  (which  is  independent  of  any  theo- 
rttical  views  respecting  the  nature  of  light ')  may  be  explained  as  follows. 
The  ray  by  which  we  see  any  object  is  not  that  which  it  emits  n^  the 
moment  we  look  at  it,  but  that  wLicb  it  did  emit  some  time  beforr.  viz.^ 
the  time  occupied  by  light  in  traversing  the  interval  which  separates  . ". 
from  us.  The  aberration  of  such  a  body  then  arising  from  tho  earth's 
velocity  must  be  applied  as  a  correction,  not  to  the  line  joining  the  earth's 
place  at  the  moment  of  observation  with  that  occupied  by  the  body  at  the 
same  moment,  but  at  that  antecedent  instant  when  the  ray  quitted  it. 
Hence  it  is  easy  to  derive  the  rule  given  by  astronomical  writers  for  the 
case  of  a  moving  object.  From  the  known  lawn  of  its  motion  and  the 
earth's,  calculate  its  apimrent  or  relative  umjidar  motion  in  the  time 
taken  hy  light  to  traverse  its  distance  from  the  earth.  This  is  the  total 
amount  of  its  apparent  disptlacemcnt.  ltd  effect  is  to  displace  the  body 
observed  in  a  direction  contrary  to  its  apparent  motion  in  the  heavens. 
And  it  is  a  compound  or  aggregate  effect  consisting  of  two  parts,  one  of 
which  is  the  aberration,  properly  so  called,  resulting  from  the  composition 
of  the  earth's  motion  with  that  of  light,  the  other  being  what  is  not 
inaptly  termed  the  Equation  of  light,  being  the  allowance  to  be  made  for 
the  time  occupied  by  the  light  in  traversing  a  variable  space. 

(38G.)  The  complete  Reduction,  as  it  is  called,  of  an  astronomical 
observation  consists  in  applying  to  the  place  of  the  observed  heavenly 
body  as  read  off  on  the  instruments  (supposed  perfect  and  in  perfect 
adjustment)  five  distinct  and  independent  corrections,  viz.  those  for 
refractiou,  parallax,  aberration,  precession,  and  nutation.  Of  these  the 
correction  for  refraction  enables  us  to  declare  what  would  have  been  the 


% 


'The  results  of  the  undulatory  and  corpuscular  theories  of  light,  in  the  matter  of 
aberration  are,  in  the  main,  the  same.  We  say  in  the  main.  There  is,  however,  a 
minute  ditference  even  of  numerical  results.  In  the  undulatory  docirine,  the  propa- 
gation of  light  takes  place  with  equal  velocity  in  all  directions,  whoilier  the  luminary 
be  at  rest  or  in  motion.  In  the  corpuscular,  with  an  excess  of  velocity  in  the  direc- 
tion of  the  motion  over  that  in  the  contrary  equal  to  twice  the  velocity  of  the  body's 
motion.  In  the  cases,  then,  of  a  body  moving  with  equal  velocity  directly  to  and  direct 
ly  from  the  earth,  the  aberration  will  be  alike  on  the  undulatory,  but  different  on  the 
corpuscular  hypothesis.  'I'he  utmost  difference  which  can  arise  from  this  cause  in  our 
system  cannot  amount  to  above  six  thousandths  of  a  second. 


182 


OUTLINES  OP   ASTRONOMY. 


observed  place,  were  there  no  atmosphere  to  displace  it.  That  for  pnral 
lax  enables  us  to  say  from  i  place  observed  at  the  surface  of  the  earth 
where  it  would  have  been  ^^ccn  if  observed  from  the  centre,  T'l.itfoi 
aberration,  where  it  would  have  been  observed  from  a  motionloaii,  uccead 
of  a  moving  station :  while  the  corrections  for  precession  and  nutation 
refer  it  to  fixed  and  determinate  instead  of  constantly  varying  celcstiiil 
circles.  The  great  importance  of  these  corrections,  which  pervade  all 
astronomy,  and  have  to  be  applied  to  every  observation  before  it  can  b' 
employed  for  any  practical  or  theoretical  purpose,  renders  thia  reoapitulu 
tion  far  from  superfluous. 

(337.)  Refraction  has  been  already  sufficiently  explained,  Art.  40,  and 
it  is  only,  therefore,  necessary  here  to  aiM  that  in  its  use  as  an  astronomi- 
cal correction  itii  amount  must  be  applied  in  a  contrary  sense  to  that  in 
which  it  affects  the  observation;  a  romtrk  equally  applicable  to  all  other 
corrections. 

(338.)  The  general  nature  of  parallax  or  rather  of  parallactic  motion 
has  also  been  explained  in  Art.  80.  But  parallax  in  the  uranographical 
sense  of  the  word  has  a  more  technical  meaning.  It  is  understood  to 
express  that  optical  displacement  of  a  body  observed  which  is  due  to  its 
being  observed,  not  from  that  point  which  we  have  fixed  upon  as  a  con- 
ventional central  stati.  n  (from  Avhich  we  conceive  the  apparent  motion 
would  be  more  simp'f  in  l*s  laws,)  but  from  some  other  station  remote 
from  such  convenfii' tij  catre:  not  from  the  centre  of  the  earth,  for 
instance,  but  from  its  siirfHce  :  not  from  the  centre  of  the  sun  (which,  as 
we  shall  hereafter  see,  '<.i  for  some  purposes  a  preferable  conventional 
station),  but  from  that  of  the  earth.  In  the  former  case  this  optical  dis- 
placement is  called  the  diurnal  or  geocentric  parallax  j  in  the  latter  the 
annnal  or  heliocentric.  In  either  case  parallax  is  the  correction  to  be 
applied  to  the  apparent  place  of  the  heavenly  body,  as  actually  seen  from 
the  station  of  observation,  to  reduce  it  to  its  place  as  it  would  have  been 
seen  at  that  instant  from  the  conventioorsl  station. 

(339.)  The  diurnal  or  geocentric  parallax  at  any  place  of  the  earth's 
surface  is  easily  calculated  if  we  know  the  dist^nice  of  the  body,  and,  vice 
versdf  if  we  know  the  diurnal  parallax  that  distance  may  be  calculated. 
For  supposing  S  the  object,  C  the  centre  of  the  earth,  A  the  station  of 
observation  at  its  surface,  and  C  A  Z  the  direction  of  a  perpendicular  to 
the  surface  at  A,  then  will  the  object  be  seen  from  A  in  the  direction  A 
S,  and  its  apparent  zenith  distance  will  be  Z  A  S ;  whereas,  if  seen  from 
the  centre,  it  will  appear  in  the  direction  C  S,  with  an  angular  distance 
from  the  zenith  of  A  equal  to  Z  C  S ;  so  that  ZAS  —  ZCSorASC 
is  the  parallax.     Now  since  by  trignometry  C  S  :  C  A  : :  sin  C  A  S 


BUMMART  or   URANOORAPIIIOAL   CORRECTIONS. 

Fig.  60, 


188 


=  sin  Z  A  S  :  sin  A  S  C,  it  follows  that  the  sine  of  the  parallax 

Radius  of  earth         .    „   .   ,, 

=-—- Tu~T  ^  sm  Z  A  S. 

Distauco  of  body 

(340.)  The  diurnal  or  geocentric  paruilax,  therefore,  at  a  given  place, 
and  for  a  given  distance  of  the  body  observed,  is  proportional  to  the  sine 
of  its  apparent  zenith  distance,  and  is,  therefore,  the  greatest  when  the 
body  is  observed  in  the  act  of  rising  or  setting,  in  which  case  its  parallax 
is  called  its  liorizontal  parallax,  so  that  at  any  other  zenith  distance, 
parallax  =  horizontal  pa:-allax  X  sine  of  apparent  zenith  distance,  and 
since  A  C  S  is  always  less  ^han  Z  A  S  It  appears  that  the  application  of 
the  reduction  or  c(,irrection  for  parallax  always  acts  in  diminution  of  the 
apparent  zenith  distance  or  increase  of  the  apparent  altitude  or  distance 
from  the  Nadir,  i.  e.  in  a  contrary  sense  to  that  for  refraction. 

(341.)  In  precisely  the  same  manner  as  the  geocentric  or  diurnal 
parallax  refers  itself  to  the  zenith  of  the  observer  for  its  direction  and 
quantitative  rule,  so  the  heliocentric  oi*  annual  parallax  refers  itself  tor  its 
law  to  the  point  in  the  heavens  diametrically  opposite  to  the  place  of  the 
sun  as  seen  from  the  earth.  Applied  as  a  correction,  its  effect  takes  place 
in  a  plane  passing  through  the  sun,  the  earth,  and  the  observed  body. 
Its  effect  is  always  to  decrease  its  observed  distance  from  that  point  or  to 
increase  its  angular  distance  from  the  sun.  And  its  sine  is  given  by  the 
relation.  Distance  of  the  observed  body  from  the  sun  :  distance  of  the 
earth  from  the  sun  : :  sine  of  apparent  angular  distance  of  the  body  from 
the  i3un  (or  its  apparent  elongation)  :  sine  of  heliocentric  parallax.' 

'  This  account  of  the  law  of  heliocentric  parallax  is  in  anticipation  of  what  follows  in 
a  subsequent  chapter,  and  will  be  better  understood  by  the  student  when  somewhat 
farther  advanced. 


I  isifilf 

mm 

i'.V>i 


\fn\ 


IMAGE  EVALUATION 
TEST  TARGET  (AAT-3) 


1.0 


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kl|28     |Z5 

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L25  11114   111.6 


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Photographic 

Sciences 

Corporation 


23  WiST  MAIN  STREET 

WEBSTER,  N.Y.  145S0 

(716)872-4503 


184 


OUTLINES  OF  ASTRONOMY.       ^kMyiV;" 


(342.)  On  a  summary  view  of  the  whole  of  the  uranographical  correc 
tious,  they  divide  themselves  into  two  classes,  those  which  do,  and  those 
which  do  not,  alter  the  apparent  configurations  of  the  heavenly  bodies 
inter  se.  The  former  are  real,  the  latter  technical  con-ections.  The  real 
corrections  are  refraction,  aberration  and  parallax.  The  technical  are  pre- 
cession and  nutation,  unless,  indeed,  we  choose  to  consider  parallax  as  a 
technical  correction  introduced  with  a  view  to  simplification  by  a  better 
choice  of  our  point  of  sight. 

(343.)  The  corrections  of  the  first  of  these  classes  have  one  peculiarity 
in  respect  of  their  law,  common  to  them  all,  which  the  student  of  prac- 
tical astronomy  will  do  well  to  fix  in  his  memory.  Thei/  all  refer  them- 
selves to  definite  apexes  or  points  of  convergence  in  the  ^here.  Thus, 
refraction  in  its  apparent  effect  causes  all  celestial  objects  to  draw  together 
or  convergb  towards  the  zenith  of  the  observer:  geocentric  parallax, 
towards  his  Nadir :  heliocentric,  towards  the  place  of  the  sun  in  the 
heavens :  aberration  towards  that  point  in  the  celestial  sphere  which  is 
the  vanishing  pomt  of  all  lines  parallel  to  the  direction  of  the  earth's 
motion  at  the  moment,  or  (as  will  be  hereafter  explained)  towards  a  point 
in  the  great  circle  called  the  ecliptic,  90°  behind  the  sun's  place  in  that 
circle.  When  applied  as  corrections  to  an  observation,  these  directions 
are  of  course  to  be  reversed. 

(344.)  In  the  quantitative  law,  too,  which  this  class  of  corrections 
follow,  a  liko  agreement  takes  place,  at  least  as  regards  the  geocentric  and 
heliocentric  parallax  and  aberration,  in  all  three  of  which  the  amount  of 
the  correction  (or  more  strictly  its  sine)  increases  in  the  direct  proportion 
of  the  sine  of  the  apparent  distance  of  the  observed  body  from  the  apex 
appropriate  to  the  particular  correction  in  question.  In  the  case  of  re- 
fraction the  law  is  less  simple,  agreeing  more  nearly  with  the  tangent  than 
the  sine  of  that  distance,  but  agreeing  with  the  others  in  placing  the 
maximum  at  90°  from  its  apex. 

(345.)  As  respects  the  order  in  which  these  corrections  are  to  be 
applied  to  any  observation,  it  is  as  follows :  1.  Refraction;  2.  Aberration; 
3.  Geocentric  Parallax;  4.  Heliocentric  Parallax;  5.  Nutation;  6.  Pre- 
cession. Such,  at  least,  is  the  order  in  theoretical  strictness.  But  as  the 
amount  of  aberration  and  nutation  is  in  all  cases  a  very  minute  Quantity, 
it  matters  not  in  what  order  they  are  applied ;  so  that  for  practical  conve- 
nience they  are  always  thrown  together  with  the  precession,  and  a;^)plied 
after  the  others. 


<v\ 


or  THE  SUN  S  MOTION. 


185 


jorrec 

those 
bodies 
be  real 
re  pre- 
X  as  a 

better 

aliarity 
)f  prac- 
r  them- 
ThuB, 
together 
)arallax, 
1  in  the 
which  is 
5  earth's 
3  a  point 
I  in  that 
[irections 

irrections 
itric  and 
aount  of 
roportion 
the  apex 
tse  of  re- 
lent than 
tcing  the 


-:-,^p!H\^.r';    iBi   f!;-':   .;te       CHAPTER  VI.  !»^^- ''"'••■  •''    '•  n -■■•...a    ,-.;iv 
.-,,f;vJ  ;Kti;i>"'  ,''iv;.\5-.j  > 7  l.ky  ,:-\  /■.>;>;-•!•;! 


1  ,. 


:'   ;o':;;.J,sl.Hyr-,:^5i.  '^i  :,1  {=',/■•: •-:■■: fi^. 


;-!:>  ;*^!-5    .'il 


«     OP    THE    SUN*S    MOTION.       '    * 


■■>:•  i     .-V; 


APPARENT  MOTION  OP  THE  SUN  NOT  UNIFORM. — ITS  APPARENT 
DIAMETER  ALSO  VARIABLE. — VARIATION  OP  ITS  DISTANCE  CON- 
CLUDED.—  ITS    APPARENT    ORBIT  AN   ELLIPSE  ABOUT  TEE    FOCUS. 

—  LAW  OP  THE  ANGULAR  VELOCITY. — EQUABLE  DESCRIPTION  OP 
AREAS.  —  PARALLAX  OP  THE  SUN.  —  ITS  DISTANCE  AND  MAGNI- 
TUDE. —  COPERNICAN  EXPLANATION  OP  THE  SUN's  APPARENT 
MOTION. — PARALLELISM   OP  THE  EARTH's  AXIS. — THE    SEASONS. 

—  HEAT  RECEIVED  FROM  THE  SUN  IN  DIFFERENT  PARTS  OP  THE 
ORBIT. — MEAN  AND  TRUE  LONGITUDES  OF  THE  SUN. — EQUATION 
OP  THE  CENTRE. — SIDEREAL,   TROPICAL,  AND  ANOMALISTIC  YEARS. 

—  PHYSICAL  CONSTITUTION   OF  THE   SUN. — ITS   SPOTS. — FACULiB. 

—  PROBABLE  NATURE  AND  CAUSE  OF  THE  SPOTS.  —  ATMOSPHERE 
OP  THE  SUN. — ITS  SUPPOSED  CLOUDS.  —  TEMPERATURE  AT  ITS 
SURFACE.  —  ITS  EXPENDITURE  OF  HEAT. — TERRESTRIAL  EFFECTS 
OF  SOLAR  RADIATION. 

(346.)  In  the  foregoing  chapters,  it  has  been  shown  that  the  apparent 
path  of  the  sun  is  a  great  circle  of  the  sphere,  which  it  performs  in  a 
period  of  one  sidereal  year.  From  this  it  follows,  that  the  line  joining 
the  earth  and  sun  lies  constantly  in  one  plane;  and  that,  therefore, 
whatever  be  the  real  motion  from  which  this  apparent  motion  arises,  it 
must  be  confined  to  one  plane,  which  is  called  the  plane  of  the  ecliptic. 

(347.)  We  have  already  seen  (art.  146)  that  the  sun's  motion  in  right 
ascension  among  the  stars  is  not  uniform.  This  b  partly  accounted  for 
by  the  obliquity  of  the  ecliptic,  in  consequence  of  which  equal  variations 
in  longitude  do  not  correspond  to  equal  changes  of  right  ascension.  But 
if  we  observe  the  place  of  the  sun  daily  throughout  the  year,  by  the 
transit  and  circle,  and  from  these  calculate  the  longitude  for  each  day,  it 
will  still  be  found  that,  even  in  its  own  proper  path,  its  apparent  angular 
motion  is  far  from  uniform.  The  change  of  longitude  in  twenty-four 
mean  solar  hours  average*  0°  59'  8''-38 ;  but  about  the  31st  of  Decern- 


186 


OUTLINES  OF  ASTKOVOVY. 


ber  it  amounts  to  1°  1'  9"-9,  and  about  the  Ist  of  July  is  only  0"  67' 
11  "-5.  Such  are  the  extreme  limits,  and  such  the  mean  value  of  the 
sun's  apparent  angular  velocity  in  its  annual  orbit. 

(348.)  This  variation  of  its  angular  velocity  is  accompanied  with  a 
corresponding  change  of  its  distance  from  us.  The  change  of  distance  is 
recognized  by  a  variation  observed  to  take  place  in  its  apparent  diameter, 
when  measured  at  different  seasons  of  the  year,  with  an  instrument 
adapted  for  that  purpose,  called  the  heliometer,^  or,  by  calculating  from 
the  time  which  its  disc  takes  to  traverse  the  meridian  in  the  transit 
instrument.  The  greatest  apparent  diameter  corresponds  to  the  1st  of 
December,  or  ij  the  greatest  angular  velocity,  and  measures  32'  35"'6, 
the  least  is  31'  31"-0;  and  corresponds  to  the  1st  of  July;  at  which 
epochs,  as  we  have  seen,  the  angular  motion  is  also  at  its  extreme  limit 
either  way.  Now,  as  we  cannot  suppose  the  sun  to  alter  its  real  size 
periodically,  the  observed  change  of  its  apparent  sizb  can  only  arise  from 
an  actual  change  of  distance.  And  the  sines  or  tangents  of  such  small 
arcs  being  proportional  to  the  arcs  themselves,  its  distances  from  us,  at 
t:'  J  above-named  epoch,  must  be  in  the  inverse  proportion  of  the  apparent 
diameters.  It  appears,  therefore,  that  the  greatest,  the  mean,  and  the 
least  distances  of  the  sun  from  us  are  in  the  respective  proportions  of  the 
numbers  101679,  1.00000,  and  0-98321 ;  and  that  its  apparent  angular 
velocity  diminishes  as  the  distance  increases,  and  vice  verad. 

(349.)  It  follows  from  this,  that  the  real  orbit  of  the  sun,  as  referred 
to  the  earth  supposed  at  rest,  is  not  a  circle  with  the  earth  in  the  centre. 
The  situation  of  the  earth  within  it  is  excentric,  '-he  excentricity  amount- 
ing to  001679  of  the  mean  distance,  which  may  '  regarded  as  our  unit 
of  measure  in  this  inquiry.  But  besides  this,  ;  'orm  of  the  orbit  is 
not  circular,  but  elliptic.  If  from  any  point  0,  uilcen  to  represent  the 
earth,  we  draw  a  line,  0  A,  in  some  fixed  direction,  from  which  we  then 


\  i 


set  off  a  series  of  angles,  A  0  B,  A  0  C,  &c.  equal  to  the  observed  longi- 
tudes of  the  sun  throughout  the  year,  and  in  these  respective  direction? 


'  'HAiof  the  sun,  and  ncrpttv  to  measure. 


I'{ 


lORM  OF  THB  SUN'S  APPARENT  ORBIT. 


18T 


measure  off  from  0  the  distances  0  A,  0  B,  0  C,  &o.  representing  tbe 
distances  deduced  from  the  observed  diameter,  and  then  connect  all  the 
dxtremities  A,  B,  0,  &o.  of  these  lines  by  a  continuous  curve,  it  is  evident 
this  will  be  a  correct  representation  of  the  relative  orbit  of  the  sun  about 
tbe  earth.  Now,  when  this  is  done,  a  deviation  from  the  circular  figure 
in  the  resulting  curve  becomes  apparent;  it  is  found  to  be  evidently  longer 
than  it  is  broad  —  that  is  to  say,  elliptic,  and  the  point  0  to  occupy,  not 
the  centre,  but  one  of  the  foci  of  the  ellipse.  The  graphical  process  here 
described  is  sufficient  to  point  out  the  general  figure  of  the  curve  in  ques- 
tion ;  but  for  the  purposes  of  exact  verification,  it  is  necessary  to  recur  to 
the  properties  of  the  ellipse,'  and  to  express  the  distance  of  any  one  of 
its  points  in  terms  of  the  angular  situation  of  that  point  with  respect  to 
the  longer  axis,  or  diameter  of  the  ellipse.  This,  however,  is  readily 
done ;  and  when  numerically  calculated,  on  the  supposition  of  the  excen- 
tricity,  being  such  as  above  stated,  a  perfect  coincidence  is  found  to 
subsist  between  the  distances  thus  computed,  and  those  derived  from  the 
measurement  of  the  apparent  diameter. 

(350.)  The  mean  distance  of  the  earth  and  sun  being  taken  for  unity, 
the  extremes  are  1'01679  and  0*98321.  But  if  we  compare,  in  like 
manner,  the  mean  or  average  angular  velocity  with  the  extremes,  greatest 
and  least,  we  shall  find  these  to  be  in  the  proportions  of  1'03386, 1-00000, 
and  0-96670.  The  variation  of  the  sun's  angular  velocity,  then,  is  much 
greater  in  proportion  than  that  of  its  distance  —  fully  twice  as  great ;  and 
if  we  examine  its  numerical  expressions  at  difterent  periods,  comparing 
them  with  the  mean  value,  and  also  with  the  corresponding  distances,  it 
will  be  found,  that,  by  whatever  fraction  of  its  mean  value  the  distance 
exceeds  the  mean,  the  angular  velocity  will  fall  short  of  its  mean  or  ave- 
rage quantity  by  very  nearly  twice  as  great  a  fraction  of  the  latter,  and 
vice  versd.  Hence  we  are  led  to  conclude  that  the  angular  velocity  is  in 
the  inverse  proportion,  not  of  the  distance  simply,  bvt  of  its  square^  eo 
that,  to  compare  the  daily  motion  in  longitude  of  the  sun,  at  one  point, 
A,  of  its  path,  with  that  at  B,  we  must  state  the  proportion  thus :  — 

0  B*  :  0  A'  : :  daily  motion  at  A  :  daily  motion  at  B.  And  this  is 
found  to  be  exactly  verified  in  every  part  of  the  orbit. 

(351.)  Hence  we  deduce  another  remarkable  conclusion  —  viz.  that  if 
the  sun  be  supposed  really  to  move  around  the  circumference  of  this 
ellipse,  its  actual  speed  cannot  be  uniform,  but  must  be  greatest  at  its 
least  distance  and  less  at  its  greatest.     For,  were  it  uniform,  the  apparent 

'  See  Conic  Sections,  by  the  Rev.  H.  P.  Hamilton,  or  any  other  of  the  very 
numerous  works  on  this  subject. 


188 


OUTLINES   OF  ASTRONOHT. 


i 


angular  velocity  would  be,  of  course,  inversely  proportional  to  the  distance ; 
simply  because  the  same  linear  change  of  place,  being  produced  in  the 
same  time  at  different  distances  from  the  eye,  must,  by  the  laws  of  per- 
spective, correspond  to  apparent  angular  displacements  inversely  as  those 
distances.  Since,  then,  observation  indicates  a  more  rapid  law  of  varia- 
tion in  the  angular  velocities,  it  is  evident  that  mere  change  of  distance, 
unaccompanied  with  a  change  of  actual  speed,  is  insufficient  to  account  for 
it ;  and  that  *he  increased  proximity  of  the  sun  to  the  earth  must  be 
accompanied  with  an  actual  increase  of  its  real  velocity  of  motion  along 
its  path.        -J  uaj^n  ><i'i  ^.i^K«J•^hf«RfTS'*•"^i*•7■("?■■  <!';;'-:>T'Sn«T  n^f^v,.sy':^jjc't 

(352.)  This  elliptic  form  of  the  sun's  path,  the  exoentrio  position  of 
the  earth  within  it,  and  the  unequal  speed  with  which  it  is  actually 
traversed  by  the  sun  itself,  all  tend  to  render  the  calculation  of  its  longi- 
tude from  theory  (i.  e.  from  a  knowledge  of  the  causes  and  nature  of  its 
motion)  difficult ;  and  indeed  impossible,  so  long  as  the  law  of  its  actual 
velocity  continues  unknown.  This  law,  however,  is  not  immediately 
apparent.  It  does  not  come  forward,  as  it  were,  and  present  itself  at  onc«, 
like  the  elliptic  form  of  the  orbit,  by  a  direct  comparison  of  angles  and 
distances,  but  requires  an  attentive  consideration  of  the  whole  series  of 
observations  registered  during  an  entire  period.  It  was  not,  therefore, 
without  much  painful  and  laborious  calculation,  that  it  was  discovered  by 
Kepler  (who  was  also  the  first  to  ascertain  the  elliptic  form  of  the  orbit), 
and  announced  in  the  following  terms :  —  Let  a  line  be  always  supposed 
to  connect  the  sun,  supposed  in  motion,  with  the  earth,  supposed  at  rest ; 
then,  as  the  sun  moves  along  its  ellipse,  this  line  (which  is  called  in  astro- 
iiomy  the  radius  vector)  will  describe  or  sweep  over  that  portion  of  the 
whole  area  or  surface  of  the  ellipse  which  is  included  between  its  consec- 
utive  positions  :  and  the  motion  of  the  sun  will  be  such  that  equal  areas 
are  thus  stcep<  over  by  the  revolving  radius  vector  in  egytal  times,  in  what- 
ever part  of  the  circumference  of  the  ellipse  the  sun  may  be  moving. 

(353.)  From  this  it  necessarily  follows,  that  in  unequal  times,  the  areas 
described  must  be  proportional  to  the  times.  Thus,  in  the  figure  of  art. 
349,  the  time  in  which  the  sun  moves  from  A  to  B,  is  to  the  time  in 
which  it  moves  from  C  to  D,  as  the  area  of  the  elliptic  sector  A  0  B  is  to 
the  area  of  the  sector  DOC. 

(354.)  The  circumstances  of  the  sun's  apparent  annual  motion  may, 
therefore,  be  summed  up  as  follows : — It  is  performed  in  an  orbit  lying  in 
one  plane  passing  through  the  earth's  centre,  called  the  plane  of  the  ecliptic, 
and  whose  projection  on  the  heavens  is  the  great  circle  so  called.  In  tbis 
plane,  however,  the  actual  path  is  not  circular,  but  elliptical ;  having  the 
earth,  not  in  its  centre,  but  in  one  focus.     The  excentrioity  of  this  ellipse 


I 


I 


DISTANOB  07  THB  SUN. 


189 


is  0'01679,  in  parts  of  a  vnit  equal  to  the  mean  diittmee,  or  7mI/  the 
longer  diameter  of  the  ellipse;  i.  e.  aboct  one  sixtieth  part  of  that  semi* 
diameter;  and  the  motion  of  the  sun  in  its  circumference  is  so  regulated, 
that  equal  areas  of  the  ellipse  are  passed  over  by  the  radius  vector  in 
equal  times.  .■i'>  <\  s,f4i  ■  .xi'-vA  -iK-r- 

(855.)  What  we  have  here  stated  supposes  no  knowledge  of  the  sun's 
actual  distance  from  the  earth,  nor,  consequently,  of  the  actual  dimen- 
sions of  its  orbit,  nor  of  the  body  of  the  sun  itself.  To  come  to  any 
conclusions  on  these  points,  we  must  first  consider  by  what  means  we  can 
arrive  at  any  knowledge  of  the  distance  of  an  object  to  which  we  have  no 
access.  Now,  it  is  obvious,  that  its  parallax  alone  can  afford  us  any  in- 
formation on  this  subject.  Suppose  P  A  B  Q  to  represent  the  earth,  0 
its  centre,  and  S  the  sun,  and  A,  B  two  situations  of  a  spectator,  or, 
which  comes  to  the  same  thing,  the  stations  of  two  spectators,  both  ob- 
serving the  sun  S  at  the  same  instant.  The  spectator  A  will  see  it  in  the 
direction  A  S  a,  and  will  refer  it  to  a  point  a  in  the  infinitely  distant 
sphere  of  the  stars,  while  the  spectator  B  will  see  it  in  the  direction  B  Si, 
and  refer  it  to  h.    The  angle  included  between  these  directions,  or  the 

Fig.  52. 


measure  of  the  celestial  arc  a  &,  by  which  it  is  dispkbced,  is  equal  to  the 
angle  A  S  B ;  and  if  this  angle  be  known,  and  the  local  situations  of  A 
and  B,  with  the  part  of  the  earth's  surface  A  B  inoludexi  between  them, 
it  is  evident  that  the  distance  C  S  may  be  calculated.  Now,  since  A  S  C 
(art.  839)  is  the  parallax  of  the  sun  as  seen  from  A,  and  B  S  C  as  seen 
from  B,  the  angle  A  S  B,  or  the  total  apparent  displacement  is  the  sum 
of  the  two  parallaxes.  Suppose,  then,  two  observers  —  one  in  the 
northern,  the  other  in  the  southern  hemisphere — at  stations  on  the  same 
meridian,  to  observe  on  the  same  day  the  meridian  altitudes  of  the  sun's 
centre.  Having  thence  derived  the  apparent  zenith  distances,  and  cleared 
them  of  the  effects  of  refraction,  if  the  distance  of  the  sun  were  equal 
to  that  of  the  fixed  stars,  the  sum  of  the  zenith  distanceiti  thus  found 
would  be  precisely  equal  to  the  sum  of  the  latitudes  north  and  south  of 
the  places  of  observation.    For  the  sum  in  question  would  then  be  equal 


190 


OUTLINES  OF  ASTRONOMT. 


to  the  angle  ZCX,  which  is  the  meridional  distance  of  the  stations 
across  the  equator.  But  the  effect  of  parallax  being  in  both  cases  to  in< 
crease  the  apparent  zenith  distances,  their  observed  sum  will  be  greater 
than  the  sum  of  the  latitudes,  by  the  sum  of  the  two  parallaxes,  or  by  the 
angle  A  S  B.  This  angle,  then,  is  obtained  by  subducting  the  sum  of 
the  north  and  south  latitudes  from  that  of  the  zenith  distances ;  and  this 
once  determined,  the  horizontal  parallax  is  easily  found,  by  dividing  the 
angle  so  determined  by  the  sum  of  the  sines  of  the  two  latitudes. 

(356.)  If  the  two  stations  be  not  exactly  on  the  same  meridian  C%  con< 
dition  very  difficult  to  fulfil),  the  same  process  will  apply,  if  we  take  care 
to  allow  for  the  change  of  the  sun's  actual  zenith  distance  in  the  interval 
of  time  elapsing  between  its  arrival  on  the  meridians  of  the  stations.  This 
change  is  readily  ascertained,  either  from  tables  of  the  sun's  motion, 
grounded  on  the  experience  of  a  long  course  of  observations,  or  by  actual 
observation  of  its  meridional  altitude  on  several  days  before  and  after  that 
on  which  the  observations  for  parallax  are  taken.  Of  course,  the  nearer 
the  stations  are  to  each  other  in  longitude,  the  less  is  this  interval  of  timfe, 
and,  consequently,  the  smaller  the  amount  of  this  correction ;  and,  there- 
fore, the  less  injurious  to  the  accuracy  of  the  final  result  is  an  v  uncertainty 
in  the  daily  change  of  zenith  distance  which  may  arise  from  imperfection 
in  the  solar  tables,  or  in  the  observations  made  to  determine  it. 

(857.)  The  horizontal  parallax  of  the  sun  has  been  concluded  from 
observations  of  the  nature  above  described,  performed  in  stations  the  most 
remote  from  each  other  in  latitude,  at  which  observatories  have  been  in- 
stituted. It  has  also  been  deduced  from  other  methods  of  a  more  refined 
nature,  and  susceptible  of  much  greater  exactness,  to  be  hereafter  de- 
scribed. Its  amount  so  obtained,  is  about  8  "'6.  Minute  as  this  quan- 
tity is,  there  can  be  no  doubt  that  it  is  a  tolerably  correct  approximation 
to  the  truth ;  and  in  conformity  with  it,  we  must  admit  the  sun  to  be 
situated  at  a  mean  distance  from  us,  of  no  less  than  23984  times  the 
length  of  the  earth's  radius,  or  about  95000000  miles. 

(358.)  That  at  so  vast  a  distance  the  sun  should  appear  to  us  of  the 
size  it  does,  and  should  so  powerfully  influence  our  condition  by  its  heat 
and  light,  requires  us  to  form  a  very  grand  conception  of  its  actual  mag- 
nitude, and  of  the  scale  on  which  those  important  processes  are  carried  on 
within  it,  by  which  it  is  enabled  to  keep  up  its  liberal  and  unceasing 
supply  of  these  elements.  As  to  its  actual  magnitude  we  can  be  at  no 
loss,  knowing  its  distance,  and  the  angles  under  which  its  diameter  appears 
to  us.  An  object,  placed  at  the  distance  of  95000000  miles,  and  sub- 
tending an  angle  of  32'  3",  must  have  a  real  diameter  of  882000  miles. 
Such,  then,  is  the  diameter  of  this  stupendous  globe.    If  we  compare  it 


"•fir 


THE  earth's  annual  MOTION. 


191 


with  what  we  have  already  ascertained  of  the  dimensions  of  our  own,  we 
shall  find  that  in  linear  magnitude  it  exceeds  the  earth  in  the  proportion 
111^  to  1,  and  in  bulk  in  that  of  1884472  to  1. 

(859.)  It  is  hardly  possible  to  avoid  associating  our  conception  of  an 
object  cf  definite  globular  figure,  and  of  such  enormous  dimensions,  with 
some  corresponding  attribute  of  mossiveness  and  material  solidity.  That 
the  sun  is  not  a  mere  phantom,  but  a  body  having  its  own  peculiar  struo- 
ture  and  economy,  our  telescopes  distinctly  inform  us.  They  show  us 
dark  spots  on  its  surface,  which  slowly  change  their  places  and  forms,  and 
by  attending  to  whose  situation,  at  different  times,  astronomers  have  ascer< 
tained  that  the  sun  revolves  about  an  axis  nearly  perpendicular  to  the 
plane  of  the  ecliptic,  performing  one  rotation  in  a  period  of  about  25  days, 
and  in  the  same  direction  with  the  diurnal  rotation  of  the  earth,  i.  e.  from 
west  to  east.  Here,  then,  we  have  an  analogy  with  our  own  globe ;  the 
slower  and  more  majestic  movement  only  corresponding  with  the  greater 
dimensions  of  the  machinery,  and  impressing  us  with  the  prevalence  of 
similar  mechanical  laws,  and  of,  at  least,  such  a  community  of  nature  as 
the  existence  of  inertia  and  obedience  to  force  may  argue.  Now,  in  the 
exact  proportion  in  which  we  invest  our  idea  of  this  immense  bulk  with 
the  attribute  of  inertia,  or  weight,  it  becomes  difficult  to  conceive  its  circu- 
lation round  so  comparatively  small  a  body  as  the  earth,  without,  on  the 
one  hand,  dragging  it  along,  and  displacing  it,  if  bound  to  it  by  some  in- 
visible tie ;  or,  on  the  other  hand,  if  not  so  held  to  it,  pursuing  its  course 
alone  in  space,  and  leaving  the  earth  behind.  If  we  connect  two  solid 
masses  by  a  rod,  and  fling  them  aloft,  we  see  them  circulate  about  a  point 
between  them,  which  is  their  common  centre  of  gravity ;  but  if  one  of 
them  be  greatly  more  ponderous  than  the  other,  this  common  ''>^ntre  will 
be  proportionally  nearer  to  that  one,  and  even  within  its  surfa  ^ :  so  that 
the  smaller  one  will  circulate,  in  fact,  about  the  larger,  witich  will  bo  com- 
paratively but  little  disturbed  from  its  place. 

(360.)  Whether  the  earth  move  round  the  sun,  the  sun  round  the 
earth,  or  both  round  their  common  centre  of  gravity,  will  make  no  dif- 
ference, so  far  as  appearances  are  concerned,  provided  the  stars  be  sup- 
posed sufficiently  distant  to  undergo  no  sensible  apparent  parallactic 
displacement  by  the  motion  so  attributed  to  the  earth.  Whether  they  are 
80  or  not  must  still  be  a  matter  of  inquiry ;  and  from  the  absence  of  any 
measurable  amount  of  such  displacement,  we  can  conclude  nothing  but 
this,  that  the  scale  of  the  sidereal  universe  is  so  great,  that  the  mutual 
orbit  of  the  earth  and  sun  may  be  regarded  as  an  imperceptible  point  in 
comparison  with  the  distance  of  its  nearest  members.  Admitting,  then, 
in  conformity  with  the  laws  of  dynamics,  that  two  bodies  connected  with 


192 


OUTLINES  or  ASTRONOMY. 


and  revolving  aboat  each  other  in  free  space  do,  in  faot,  revolve  about  their 
common  centre  of  gravity,  which  remains  immoveable  by  their  mutual 
action,  it  becomes  a  matter  of  further  inquiry,  vihereahouU  between  them 
this  centre  is  situated.  Mechanics  teach  us  that  ita  place  will  divide  their 
mutual  distance  in  the  inverse  ratio  of  their  tce^hu  or  maaaet;^  and 
calculations  grounded  on  phenomena,  of  which  an  account  will  be  given 
further  on,  inform  us  that  this  ratio,  in  the  case  of  the  sun  and  earth,  is 
actually  that  of  854936  to  1,  —  the  sun  being,  in  that  proportion,  more 
ponderous  than  the  earth.  From  this  it  will  follow  that  the  common  point 
about  which  they  both  circulate  is  only  267  miles  from  the  sun's  centre, 
or  about  ^Von^^  P^'*^  °^  ^^  ^^"^  diameter. 

(361.)  Henceforward,  then,  in  conformity  with  the  above  statements, 
and  with  the  Copemioan  view  of  our  system,  we  must  learn  to  look  upon 
the  sun  as  the  comparatively  motionless  centre  about  which  the  earth  per- 
forms an  annual  elliptic  orbit  of  the  dimensions  and  ezcentricity,  and  with 
a  velocity,  regulated  according  to  the  law  above  assigned ;  the  sun  occu- 
pying one  of  the  foci  of  the  ellipse,  and  from  that  station  quietly  dissen^i- 
nating  on  all  sides  its  light  and  heat ;  while  the  earth  travelling  round  it, 
and  presenting  itself  differently  to  it  at  different  times  of  the  year  and 
day,  passes  through  the  varieties  of  day  and  night,  summer  and  winter, 
which  we  enjoy. 

(362.)  In  this  annual  motion  of  the  earth,  its  axis  preserves,  at  all 
times,  the  same  direction  as  if  the  orbitual  movement  had  no  existence ', 
and  is  carried  round  parallel  to  itself,  and  pointing  always  to  the  same 


\ 

vanishing  point  in  the  sphere  of  the  fixed  stars.  This  it  is  which  gives 
rise  to  the  variety  of  seasons,  as  we  shall  now  explain.  In  so  doing,  we 
shall  neglect  (for  a  reason  which  will  be  presently  explained)  the  ellipticity 
of  the  orbit,  and  suppose  it  a  circle,  with  the  sun  in  the  centre. 

'  Principis,  lib.  i.  Ux.  iii.  cor.  14.  :, 


OF  THE   SEASONS. 


198 


(868.)  Let,  then,  S  represent  the  sun,  and  A,  B,  C,  D,  four  positions 
of  the  earth  in  its  orhit  90°  apart,  viz.  A  that  which  it  has  on  the  iilst 
of  March,  or  at  the  time  of  the  vernal  equinox;  B  that  of  the  2l8t  of 
June,  or  the  summer  solstice;  C  that  of  the  21st  of  September,  or  the 
autumnal  equinox;  and  D  that  of  the  21st  of  December,  or  the  winter 
solstice.  In  each  of  these  positions  let  PQ  represent  the  axis  of  the 
earth,  about  which  its  diurnal  rotation  is  performed  without  interfering 
with  its  annual  motion  in  its  orbit.  Then,  since  the  sun  can  only  en- 
lighten one  half  of  the  surface  at  once,  viz.  that  turned  towards  it,  the 
shaded  portions  of  the  globe  in  its  several  positions  will  represent  the 
dark,  and  the  bright,  the  enlightened  halves  of  the  earth's  surface  in  these 
positions.  Now,  1st,  in  the  position  A,  the  sun  is  vertically  over  the 
intersection  of  the  equinoctial  F  E  and  the  ecliptic  H  G.  It  is,  therefore, 
in  the  equinox ;  and  in  this  position  the  poles  P  Q,  both  fall  on  the  ex- 
treme confines  of  the  enlightened  side.  In  this  position,  therefore,  it  is 
day  over  half  the  northern  and  half  the  southern  hemisphere  at  once ; 
and  as  the  earth  revolves  on  its  axis,  every  point  of  its  surface  describes 
half  its  diurnal  course  in  light,  and  half  in  darkness ;  in  other  words,  the 
duration  of  day  and  night  is  here  equal  over  the  whole  globe :  hence  the 
term  equinox.  The  same  holds  good  at  the  autumnal  equinox  on  the 
position  C.  ' 

(364.)  B  is  the  position  of  the  earth  at  the  time  of  the  northern 
summer  solstice.  Here  the  north  pole  P,  and  a  considerable  portion  of 
the  earth's  surface  in  its  neighbourhood,  as  far  as  B,  are  situated  within 
the  enlightened  half.  As  the  earth  turns  on  its  axis  in  this  position, 
therefore,  the  whole  of  that  part  remains  constantly  enlightened ;  there- 
fore, at  this  point  of  its  orbit,  or  at  this  season  of  the  year,  it  is  continual 
day  at  the  north  pole,  and  in  all  that  region  of  the  earth  which  encircles 
this  pole  as  far  as  B  —  that  is,  to  the  distance  of  23**  28'  from  the  pole, 
or  within  what  is  called  in  geogrcphy,  the  arctic  circle.  On  the  other 
hand,  the  opposite  or  south  pole  Q,  with  all  the  region  comprised  within 
the  antarctic  circle,  as  far  as  23°  28'  from  the  south  pole,  are  immersed 
at  this  season  in  darkness  during  the  entire  diurnal  rotation,  so  that  it  is 
here  continual  night. 

(365.)  With  regard  to  that  portion  of  the  surface  comprehended 
between  the  arctic  and  antarctic  circles,  it  is  no  less  evident  that  the 
nearer  any  point  is  to  the  north  pole,  the  larger  will  be  the  portion  of 
its  diurnal  course  comprised  within  the  bright,  and  the  smaller  within 
the  dark  hemisphere ;  that  is  to  say,  the  longer  will  be  its  day,  and  the 
shorter  its  night.  Every  station  north  of  the  equator  will  have  a  day  of 
more  and  a  night  of  less  than  twelve  hours'  duration,  and  vice  "erifl. 
13 


194 


OUTLINES   OF  ASTRONOMY. 


All  theso  phenomena  are  exactly  inverted  when  the  earth  comes  to  the 
oppoHite  point  D  of  iti  orbit. 

(300.)  Now,  the  temperature  of  any  part  of  the  earth's  surface 
depends  mainly  on  its  exposure  to  the  sun's  rays.  Whenever  the  sun  is 
above  the  horizon  of  any  place,  that  pluco  is  receiving  heat ;  when  below, 
parting  with  it,  by  the  process  called  radiation ;  and  the  whole  quantities 
received  and  parted  with  in  the  year  (secondary  causes  apart)  must 
balance  each  other  at  every  station,  or  the  equilibrium  of  temperature 
(that  is  to  say,  the  constancy  which  is  observed  to  prevail  in  the  annual 
avernges  of  temperature  as  indicated  by  the  thermometer)  would  not  be 
supported.  Whenever,  then,  the  sun  remains  more  than  twelve  hours 
above  the  horizon  of  any  place,  and  less  beneath,  the  general  temperature 
of  that  place  will  be  above  the  average ;  when  the  reverse,  below.  As 
the  earth,  then,  moves  from  A  to  B,  the  days  growing  longer,  and  the 
nights  shorter,  in  th$  northern  hemisphere,  the  temperature  of  every  part 
of  that  hemisphere  increases,  and  we  pass  from  spring  to  summer ;  while, 
at  the  same  time,  the  reverse  obtains  in  the  southern  hemisphere.  <  As 
the  earth  passes  from  B  to  C,  the  days  and  nights  again  approach  to 
equality  —  the  excess  of  temperature  in  the  northern  hemisphere  above 
the  mean  state  grows  less,  as  well  as  its  defect  in  the  southern ;  and  at 
the  autumnal  equinox  C,  the  mean  state  is  once  more  attained.  From 
thence  to  D,  and,  finally,  round  again  to  A,  all  the  same  phenomena,  it 
is  obvious,  must  again  occur,  but  reversed,  —  it  being  now  winter  in  the 
northern  and  summer  in  the  southern  hemisphere. 

(867.)  All  this  b  exactly  consonant  to  observed  fact.  The  continual 
day  within  the  polar  circles  in  summer,  and  night  in  winter,  the  general 
increase  of  temperature  and  length  of  day  as  the  sun  approaches  the 
elevated  pole,  and  the  reversal  of  the  seasons  in  the  northern  and  southern 
hemispheres,  are  all  facts  too  well  known  to  require  further  comment. 
The  positions  A,  G  of  the  earth  correspond,  as  we  have  said,  to  the 
equinoxes ;  those  at  B,  D  to  the  tohtices.  This  term  must  be  explained. 
If,  at  any  point,  X,  of  the  orbit,  we  draw  X  P  the  earth's  axis,  and  X  S 
to  the  sun,  it  is  evident  that  the  angle  P  X  S  will  be  the  sun's  polar 
distance.  Now,  this  angle  is  at  its  maximum  in  the  position  D,  and  at 
its  minimum  at  B;  being  in  the  former  case=90°+23''  28'=103<'  28', 
and  in  the  latter  90**— 23"*  28=66°  32'.  At  these  points  the  sun 
ceases  to  approach  to  or  to  recede  from  the  pole,  and  hence  the  name 
solstice. 

(368.)  The  elliptic  form  of  the  earth's  orbit  has  but  a  very  trifling 
share  in  producing  the  variation  of  temperature  corresponding  to  the 
difference  of  seasons     This  assertion  may  at  first  sight  seem  incompati- 


EQUAL  SUPFLT  OF  UEAT  TO   BOTH   HEMISPHERES.         195 

blo  with  what  wo  know  of  tbo  laws  of  tbo  oommuDication  of  boat  from 
a  luminary  placed  at  a  variablo  distance.  Hoat,  like  ligbt,  being  equally 
dispersed  from  tbo  sun  in  all  directions,  and  being  spropH  over  tbe  surfaco 
of  a  spbere  continually  enlarging  as  wo  recede  from  tiiu  centre,  must,  of 
course,  diminisb  in  intensity  according  to  tbe  inverse  proportion  of  tbo 
surface  of  the  sphere  over  which  it  is  spread ;  that  is,  in  the  inverse  pro- 
portion of  tbe  square  of  the  distance.  But  we  have  seen  (art.  850)  that 
this  is  also  tbe  proportion  in  which  the  angular  velocity  of  the  earth 
about  the  sun  varies.  Hence  it  appears,  that  tbe  momentary  supply  of 
heat  received  by  tbe  earth  from  tbe  sun  varies  in  the  exact  proportion  of 
angular  velocity,  i.  o.  of  the  momentary  increase  of  longitude :  and  from 
ibis  it  follows,  that  equal  amounts  of  beat  are  received  from  tbe  sun  in 
passing  over  equal  angles  round  it,  in  whatever  part  of  the  ellipse  those 
angles  may  be  situated.     Let,  then,  S  represent  the  sun ;  A  Q  M  P  the 


i 


earth's  orbit;  A  its  nearest  point  to  the  sun,  or,  as  it  is  called,  tbo  peri- 
helion of  its  orbit ;  M  tbe  farthest,  or  tbe  aphelion ;  and  therefore  A  S 
M  the  axi»  of  tbe  ellipse.  Now,  suppose  the  orbit  divided  into  two 
segments  by  a  straight  line  P  S  Q,  drawn  through  the  sun,  and  anyhow 
situated  as  to  direction :  then,  if  we  suppose  the  earth  to  circulate  in  tbe 
direction  P  A  Q  M  P,  it  will  have  passed  over  180"  of  longitude  in 
moving  from  P  to  Q,  and  as  many  in  moving  from  Q  to  P.  It  appears, 
therefore,  from  what  has  been  shown,  that  the  supplies  of  heat  received 
from  the  sun  will  be  equal  in  the  two  segments,  in  whatever  direction  tbe 
line  P  S  Q  be  drawn.  They  will,  indeed,  be  described  in  unequal  times ; 
that  in  which  tbe  perihelion  A  lies  in  a  shorter,  and  the  other  in  a  longer, 
in  proportion  to  their  unequal  area :  but  the  greater  proximity  of  tbe  sun 
in  the  smaller  segment  compensates  exactly  for  its  more  rapid  description, 
and  thus  an  equilibrium  of  beat  is,  as  it  were,  maintained.  Were  it  not 
for  this,  the  excentricity  of  the  orbit  would  materially  influence  tbe  tran- 
sition of  seasons.    The  fluctuation  of  distance  amounts  to  nearly  ^^^th  of 


196 


OUTLINES  OF  ASTRONOMY. 


I 


its  mean  quantity,  and,  consequently,  the  fluctuation  in  the  sun's  direct 
heating  power  to  double  this,  or  Jgth  of  the  whole.  Now,  the  perihelion 
of  the  orbit  is  situated  nearly  at  the  place  of  the  northern  winter  solstice ; 
so  that,  were  it  not  for  the  compensation  we  have  just  described,  the  effect 
would  be  to  exaggerate  the  difference  of  summer  and  winter  in  the 
southern  hemisphere,  and  to  moderate  it  in  the  northern ;  thus  producing 
a  more  violent  alternation  of  climate  in  the  one  hemisphere,  and  an 
approach  to  perpetual  spring  in  the  other.  As  it  is,  however,  no  such 
inequality  subsists,  but  an  equal  and  impartial  distribution  of  heat  and 
light  is  accorded  to  both. 

(369.)  This  does  not  prevent,  however,  the  direct  impression  of  the 
solar  heat  in  the  height  of  summer,  —  the  glow  and  ardour  of  his  rays, 
under  a  perfectly  clear  sky,  at  noon,  in  equal  latitudes  and  under  equal 
circumstances  of  .exposure, —  from  being  very  materially  greater  in  the 
southern  hemisphere  than  in  the  northern.  One  fifteenth  is  too  considera- 
ble a  fraction  of  the  whole  intensity  of  sunshine  not  to  aggravate  in  a 
serious  degree  the  sufferings  of  those  who  are  exposed  to  it  in  thirst/ 
deserts,  without  shelter.  The  accounts  of  these  sufferings  in  the  interior 
of  Australia,  for  instance,  are  of  the  most  frightful  kind,  and  would  seem 
far  to  exceed  what  have  ever  been  undergone  by  travellers  in  the  northcra 
deserts  of  Africa.' 

(370.)  A  conclusion  of  a  very  remarkable  kind,  recently  drawn  by  Pro- 
fessor Dove  from  the  comparison  of  the  thermometric  observations  at 
different  seasons  in  very  remote  regions  of  the  globe,  may  appear  on  first 
sight  at  variance  with  what  is  above  stated.  That  eminent  uieteorologist 
has  shown,  by  taking  at  all  seasons  the  mean  of  the  temperatures  of  points 
diametrically  opposite  to  each  other,  that  the  mean  temperature  of  (lie 
whole  earth's  surface  in  June  considerably  exceeds  that  in  December. 
This  result,  which  is  at  variance  with  the  greater  proximity  of  the  sun  in 
December,  is,  however,  due  to  a  totally  different  and  very  powerful  cause, 
—  the  greater  amount  of  land  in  that  hemisphere  which  has  its  summer 
solstice  in  June  (i.  e.  the  northern,  sec  art.  362) ;  and  the  fact  is  so 
explained  by  him.  The  effect  of  land  under  sunshine  is  to  throw  heat 
into  the  general  atmosphere,  and  so  distribute  it  by  the  carrying  power  of 
the  latter  over  the  whole  earth.  Water  is  much  less  effective  iu  this 
respect,  the  heat  penetrating  its  depths,  and  being  there  absorbed  ;  so  that 

'  See  the  account  of  Captain  Sturt's  exploration  in  Athenaeum,  No.  1012.  "The 
ground  was  almost  a  molten  surface,  and  if  a  match  accidentally  fell  upon  it.  it  imme- 
diately ignited."  The  author  has  observed  the  temperature  of  the  surface  soil  ia 
South  Africa  as  high  as  159°  Fahrenheit.  An  ordinary  lucifer  match  does  not  i^'iiite 
when  simply  pressed  upon  a  smooth  surface  at  212°,  but  m  the  act  of  withdrawing  U, 
it  takes  fire,  andthe  slightest  friction  upon  such  a  surface  of  course  ignites  it. 


MEAN  TEMPERATURE   OF   THE   EARTH'S   SURFACE.  197 


tho  surface  never  acquires  a  very  elevated  temperature  even  under  the 
equator, 

(371.)  The  great  key  to  simplicity  of  conception  in  astronomy,  and, 
indeed,  in  all  sciences  where  motion  is  concerned,  consists  in  contempla- 
ting every  movement  as  referred  to  points  which  are  either  permanently 
fixed,  or  so  ^^^"'•ly  so,  as  that  their  motions  shall  be  too  small  to  interfere 
materially  with  and  confuse  our  notions.  In  the  choice  of  these  primary 
points  of  reference,  too,  we  must  endeavour,  as  far  as  possible,  to  select 
such  as  have  simple  and  symmeti'ical  geometrical  relations  of  situation  with 
respect  to  the  curves  described  by  the  moving  parts  of  the  system,  and 
which  are  thereby  fitted  to  perform  the  office  of  natural  centres  —  advan- 
tageous stations  for  the  eye  of  reason  and  theory.  Having  learned  to 
attribute  an  orbitual  motion  to  the  earth,  it  loses  this  advantage,  which  is 
transferred  to  the  sun,  as  the  fixed  centre  about  which  its  orbit  is  per- 
formed. Precisely  as,  when  embarrassed  by  the  earth's  diurnal  motion, 
we  have  learned  to  transfer,  in  imagination,  our  station  of  observation 
from  its  surface  to  its  centre,  by  the  application  of  the  diurnal  parallax ; 
so,  when  we  come  to  inquire  into  the  movements  of  the  planets,  we  shall 
find  ourselves  continually  embarrassed  by  the  orbitual  motion  of  our  point 
of  view,  unless,  by  the  consideration  of  the  annual  or  heliocentric  paral- 
lax, we  consent  to  refer  all  our  observations  on  them  to  the  centre  of  the 
sun,  or  rather  to  the  common  centre  of  gravity  of  the  sun,  and  the  other 
bodies  which  are  connected  with  it  in  our  sj'stem.  Hence  arises  the  dis- 
tinction between  the  geocentric  and  heliocentric  place  of  an  object.  The 
former  refers  its  situation  in  space  to  an  imaginary  sphere  of  infinite 
radius,  having  the  centre  of  the  earth  for  its  centre  —  the  lattet  to  one 
concentric  with  the  sun.  Thus,  when  we  speak  of  the  heliocentric  longi- 
tudes and  latitudes  of  objects,  we  suppose  the  spectator  situated  in  the  sun 
and  referring  them  by  circles  perpendicular  to  the  plane  of  the  ecliptic, 
to  the  great  circle  marked  out  in  the  heavens  by  the  infinite  prolongation 
of  that  plane. 

(372.)  The  point  in  the  imaginary  concave  of  an  infinite  heaven,  to 
which  a  spectator  in  the  sun  refers  the  earth,  must,  of  course,  be  diame- 
trically opposite  to  that  to  which  a  spectator  on  the  earth  refers  the  sun's 
centre;  consequently  the  heliocentric  latitude  of  the  earth  is  always 
nothing,  and  its  heliocentric  longitude  always  equal  to  the  suit's  geocentric 
longitude -\-lSO°.  The  heliocentric  equinoxes  and  solstices  are,  therefore, 
the  same  as  the  geocentric  reversely  named ;  and  to  conceive  them,  we 
have  only  to  imagine  a  plane  passing  through  the  sun's  centre,  parallel  to 
the  earth's  equator,  and  prolonged  infinitely  on  all  sides.   The  line  of  inter- 


1 , 


I  %i 


198 


OUTLINES   OF  ASTRONOMT. 


i^ 


! 


i 


section  of  this  plane  and  the  plane  of  the  ecliptic  is  the  line  of  equinoxes, 
and  the  solstices  are  90"  distant  from  it. 

(373.)  The  position  of  the  longer  axis  of  the  earth's  orbit  is  a  point 
of  great  importance.  In  the  figure  (art.  368)  let  E  C  L I  be  the  ecliptic, 
E  the  vernal  equinox,  L  the  autumnal  (i.  e.  the  points  to  which  the  earth 
is  re/erred  from  the  sun  when  its  heliocentric  longitudes  are  0°  and  180° 
respectiveli/).  Supposing  the  earth's  motion  to  be  performed  in  the  direc- 
tion E  C  L  I,  the  angle  E  S  A,  or  the  longitude  of  the  perihelion,  in  the 
year  1800  was  99°  30'  5" :  we  say  in  the  year  1800,  because,  in  point  of 
fact,  by  the  operation  of  causes  herea'ter  to  be  explained,  its  position  is 
subject  to  an  extremely  slow  variation  of  about  12"  per  annum  to  the 
eastward,  and  which  in  the  progress  of  an  immensely  long  period — of  no 
less  than  20984  years  —  carries  the  axis  A  S  M  of  the  orbit  completely 
round  the  whole  circumference  of  the  ecliptic.  But  this  motion  must  be 
disregarded  for  the  present,  as  well  as  many  other  minute  deviations,  to  be 
brought  into  view  when  they  can  be  better  understood. 

(374.)  Were  the  earth's  orbit  a  circle,  described  with  a  uniform 
velocity  about  the  sun  placed  in  its  centre,  nothing  could  be  easier  than 
to  calculate  its  position  at  any  time  with  respect  to  the  line  of  equinoxes, 
or  its  longitude,  for  we  should  only  have  to  reduce  to  numbers  the  pro- 
portion following;  viz.  One  year  :  the  time  elapsed  : :  360°  :  the  arc  of 
longitude  passed  over.  The  longitude  so  calculated  is  called  in  astronomy 
the  mean  longitnde  of  the  earth.  But  since  the  earth's  orbit  is  neither 
circular,  nor  uniformly  described,  this  rule  will  not  give  us  the  true  place 
in  the  orbit  of  any  proposed  moment.  Nevertheless,  as  the  excentricity 
and  deviation  from  a  circle  are  small,  the  time  place  will  never  deviate 
very  far  from  that  so  determined  (which  for  distinction's  sake,  is  called 
the  mean  place),  and  the  former  may  at  all  times  be  calculated  from  the 
latter,  by  applying  to  it  a  correction  or  equation  (as  it  is  termed),  whose 
amount  is  never  very  great,  and  whose  computation  is  a  question  of  pure 
geometry,  depending  on  the  equable  description  of  areas  by  the  earlh 
about  the  sun.  For  since,  in  elliptic  motion  according  to  Kepler's  law 
above  stated,  areas  not  angles  are  described  uniformly,  the  proportion 
must  now  be  stated  thus ; — One  year  :  the  time  elapsed  : :  the  whole  area 
of  the  ellipse  :  the  area  of  the  sector  swept  over  by  the  radius  vector  in 
that  time.  This  area,  therefore,  becomes  known,  and  it  is  then,  as  above 
observed,  a  problem  of  pure  geometry  to  ascertain  the  angle  about  the  sun 
(A  S  V,Jig.  art.  d08),  which  corresponds  to  any  proposed  fractional  area  of 
the  whole  ellipse  supposed  to  be  contained  in  the  sector  A  P  S.  Suppose 
we  set  out  from  A  the  perihelion,  then  will  the  angle  ASP  at  first 
increase  more  rapidly  than  the  mean  longitude,  and  will,  therefore,  during 


OF  THE   STJN  S   MEAN   AND   TRUE   LONGITUDES. 


199 


the  wbole  semi-rcT  ion  from  A  to  M,  exceed  it  in  amount ;  or,  in  other 
words,  the  true  place  will  be  in  advance  of  the  mean :  at  M,  one  half  the 
yoar  will  have  elapsed,  and  one  half  the  orbit  have  been  described, 
whether  it  be  circular  or  elliptic.  Here,  then,  the  mean  and  true  places 
coincide ;  but  in  all  the  other  half  of  the  orbit,  from  M  to  A,  the  true 
place  will  fall  short  of  the  mean,  since  at  M  the  angular  motion  is  slowest, 
and  the  true  place  from  this  point  begins  to  lag  behind  the  mean  —  to 
make  up  with  it,  however,  as  it  approaches  A,  where  it  once  more  over- 
takes it. 

(375.)  The  quantity  by  which  the  tme  longitude  of  the  earth  differs 
from  the  Tnean  longitude  is  called  the  equation  of  the  centre,  and  is  addi- 
tive during  all  the  half-year,  in  which  the  earth  passes  from  A  to  M, 
beginning  at  0°  0'  0",  increasing  to  a  maximum,  and  again  diminishing 
to  zero  at  M ;  after  which  it  becomes  subtractive,  attains  a  maximum  of 
subtractive  magnitude  between  M  and  A,  and  again  diminishes  to  0  at  A. 
Its  maximum,  both  additive  and  subtractive,  is  1*  55'  33"-3. 

(376.)  By  applying,  then,  to  the  earth's  mean  longitude,  the  equation 
of  the  centre  corresponding  to  any  given  time  at  which  we  would  ascer- 
tain its  place,  the  true  longitude  becomes  known ;  and  since  the  sun  is 
always  seen  from  the  earth  in  180**  more  longitude  than  the  earth  from 
the  sun,  in  this  way  also  the  sun's  true  place  in  the  ecliptic  becomes 
known.  The  calculation  of  the  equation  of  the  centre  is  performed  by  a 
table  constructed  for  that  purpose,  to  be  found  in  all  "Solar  Tables." 

(377.)  The  maximum  value  of  the  equation  of  the  centre  depends  only 
on  the  ellipticity  of  the  orbit,  and  may  be  expressed  in  terms  of  the  ex- 
centricity.  Vice  versd,  therefore,  if  the  former  quantity  can  be  ascer- 
tained by  observation,  the  latter  may  be  derived  from  it ;  because,  when- 
ever the  law,  or  numerical  connection,  between  two  quantities  is  known, 
the  one  can  always  be  determined  from  the  other.  Now,  by  assiduous 
observation  of  the  sun's  transits  over  the  meridian,  we  cart  ascertain,  for 
every  day,  its  exact  right  ascension,  and  thence  concli'.de  its  longitude 
(art.  309).  After  this,  it  is  easy  to  assign  the  angle  by  which  this 
observed  longitude  exceeds  or  falls  short  of  the  mean ;  and  the  greatest 
amount  of  this  excess  or  defect  which  occurs  in  the  whole  year,  is  the 
maximum  equation  of  the  centre.  This,  as  a  means  of  ascertaining  the 
escentricity  of  the  orbit,  is  a  far  more  easy  and  accurate  method  than 
that  of  concluding  the  sun's  distance  by  measuring  its  apparent  diameter. 
The  results  of  the  two  methods  coincide,  however,  perfectly. 

(378.)  If  the  ecliptic  coincided  with  the  equinoctial,  the  effect  of  the 
equation  of  the  centre,  by  disturbing  the  uniformity  of  the  sun's  apparent, 
motion  in  longitude,  would  cause  an  inequality  in  its  time  of  coming  on 


200 


OUTLINES   OF  ASTRONOMY. 


tho  meridian  on  successive  days.  When  the  sun's  centre  comes  to  tba 
meridian,  it  is  ajtpiirent  noon,  and  if  its  motion  in  longitude  were  uni- 
form, and  the  ecliptic  coincident  with  the  equinoctial,  this  would  always 
coincide  with  mean  noon,  or  the  stroke  of  12  on  a  well-regulated  solar 
clock.  But,  independent  of  the  want  of  uniformity  in  its  motion,  the 
obliquity  of  the  ecliptic  gives  rise  to  another  inequality  in  this  respect ;  in 
consequence  of  which,  tho  sun,  even  supposing  its  motion  in  the  ecliptic 
uniform,  would  yet  alternately,  in  its  time  of  attaining  the  meridian,  anti- 
cipate and  fall  short  of  the  mean  noon  as  shown  by  the  clock.  For  the 
right  ascension  of  a  celestial  object  forming  a  side  of  a  right-angled  sphe- 
rical triangle,  of  which  its  longitude  is  the  hypothenuse,  it  is  clear  that 
the  uniform  increase  of  the  latter  must  necessitate  a  deviation  from  uni- 
formity id  the  increase  of  the  former. 

(379.)  These  two  causes,  then,  acting  conjointly,  produce,  in  fact,  a 
veiy  considerable  fluctuation  in  the  time  as  shown  per  clock,  when  the  sun 
really  attains  the  meridian.  It  amounts,  in  fact,  to  upwards  of  half  an 
hour;  apparent  noon  sometimes  taking  place  as  much  as  10 J  min.  before 
mean  noon,  and  at  others  as  much  as  14^  min.  after.  This  dificrencc 
between  apparent  and  mean  noon  is  called  the  equation  of  time,  and  is 
calculated  and  inserted  in  ephemerides  for  every  day  of  the  year,  under 
that  title :  or  else,  which  comes  to  the  same  thing,  the  moment,  in  mean 
time,  of  the  sun's  culmination  for  each  day,  is  set  down  as  an  astrono- 
mical phaenomenon  to  be  observed. 

(380.)  As  the  sun,  in  its  apparent  annual  course,  is  carried  along  the 
ecliptic,  its  declination  is  continually  varying  between  the  extreme  limits 
of  23°  27'  30"  north,  and  as  much  south,  which  it  attains  at  the  sol- 
stices. It  is  consequently  always  vertical  over  some  part  or  other  of  that 
zone  or  belt  of  the  earth's  surface  which  lies  between  the  north  and  south 
parallels  of  23°  27'  30".  These  parallels  are  called  in  geography  the 
tropics ;  the  northern  one  that  of  Cancer,  and  the  southern,  of  Capri- 
corn ;  because  the  sun,  at  the  respective  solstices,  is  situated  iu  the  divi- 
sions, or  signs  of  the  ecliptic  so  denominated.  Of  these  signs  there  are 
twelve,  each  occupying  30°  of  its  circumference.  They  commence  at  tlio 
vernal  equinox,  and  are  named  in  order — Aries,  Taurus,  Gemini,  Cancer, 
Leo,  Virgo,  Libra,  Scorpio,  Sagittarius,  Capricornus,  Aquarius,  I*isces.' 
They  are  denoted  also  by  the  following  symbols  :  —  T,  y.  n,  s,  ^,  n_g,  £=, 
^.y  /i  y?)  '^1  X-  Longitude  itself  is  also  divided  into  signs,  degrees,  and 
minutes,  &c.     Thus  5»  27°  0'  corresponds  to  177°  0'. 

'  They  may  be  remembered  by  the  following  memorial  hexameters :  — 
Sunt  Aries,  Taurus,  Gemini,  Cancer,  Leo,  Virgo, 
Libraque,  Scorpius,  .\rcitenens,  Caper,  Amphora,  Pisces. 


SIDEREAL,   TROPICAL,  AND  ANOMALISTIC  YEARS. 


201 


(381.)  These  Siyns  are  purely  techuical  subdivisions  of  the  ecliptic, 
commencing  from  the  actual  equinox,  and  are  not  to  be  confounded  with 
the  constellations  so  called  (and  sometimes  so  symbolized).  The  constel- 
lations of  the  zodiac,  as  they  now  stand  arranged  on  the  ecliptic,  are  all  a 
full  "sign"  in  advance- or  anticipation  of  their  symbolic  cognomens  thereon 
marked.  Thus  the  constellation  Aries  actually  occupies  the  sign  Taurus 
}i,  the  constellation  Taurus,  the  sign  Gemini  n,  and  so  on,  the  supis 
Laving  retreated'  among  the  stars  (together  with  the  equinox  their  origin), 
by  the  effect  of  precession.  The  bright  star  Spica  in  the  constellation 
Virgo  (tt  Virginis),  by  the  observations  of  Hipparchus,  128  years  B.  C, 
jD'cccdcd,  or  was  westward  of  the  autumnal  equinox  in  longitude  by  6°. 
In  1750  it  followed  or  stood  eastward  of  the  same  equinox  by  20°  21'. 
Its  place  then,  as  referred  to  the  ecliptic  at  the  former  epoch,  would  be  in 
longitude  5'  21**  0',  or  in  the  24th  degree  of  the  siff'i  SI,  whereas  in  the 
latter  epoch  it  stood  in  the  21st  degree  of  W,  the  equinox  having  retreated 
by  26°  21'  in  the  interval,  1878  years,  elapsed.  To  avoid  this  source  of 
misunderstanding,  the  use  of  "  signs"  and  their  symbols  in  the  reckoning 
of  celestial  longitudes  is  now  almost  entirely  abandoned,  and  the  ordinary 
reckoning  (by  degrees,  &c.  from  0  to  360)  adopted  in  its  place,  and  the 
names  Aries,  Virgo,  &c.  are  becoming  restricted  to  the  constellations  so 
called.^ 

(382.)  IVhen  the  sun  is  in  either  tropic,  it  ealighteas,  as  wo  have  seen, 
the  pole  on  that  side  the  equator,  and  shines  over  or  beyond  it  to  the 
extent  of  23°  27'  30".  The  parallels  of  latitude,  at  this  distance  from 
either  pole,  are  called  the  polar  circles,  and  are  distinguished  from  each 
other  by  the  names  arctic  and  antarctic.  The  regions  within  these 
circles  are  sometimes  termed  frigid  zones,  while  the  belt  between  .he 
tropics  is  called  the  torrid  zone,  and  the  intermediate  belts  temperate  zones. 
These  last,  however,  are  merely  names  given  for  the  sake  of  naming ;  as, 
in  fact,  owino;  to  the  different  distribution  of  land  and  sea  in  the  two  hemi- 
spheres,  zones  of  climate  are  not  co-terminal  with  zones  of  latitude. 

(383.)  Our  seasons  are  determined  by  the  apparent  passages  of  the  sun 
across  the  equinoctial,  and  its  alternate  arrival  in  the  northern  and  south- 
ern hemisphere.  Were  the  equinox  invariable,  this  would  happen  at 
intervals  precisely  equal  to  the  duration  of  the  sidereal  year ;  but,  in  fact, 

'  Retreated  is  here  used  with  reference  to  lo7igitude,  not  to  the  apparent  diurnal 
motion. 

"  When,  however,  the  place  of  the  sun  is  spoken  of,  the  old  usage  prevails.  Thus, 
if  we  say  "  the  sun  is  in  Aries,"  it  would  be  interpreted  to  mean  between  0°  and  30' 
of  longitude.  So,  also,  "the  first  point  of  Aries"  is  still  understood  to  mean  the 
vernal,  and  "  the  first  point  of  Libra,"  the  autumnal  equinox;  and  so  in  a  few  other 
cases. 


r1: 

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202 


OUTLINES   OP  ASTRONOMY. 


I 


owing  to  tlie  slow  conical  motion  of  the  earth's  axis  described  in  art.  317, 
the  equinox  retreats  on  the  ecliptic,  and  meets  the  advancing  sun  some- 
what I)f/ore  the  whole  sidereal  circuit  is  completed.  The  annual  retreat 
of  the  equinox  is  50"'l,  and  this  arc  is  described  by  the  sun  in  the  eclip- 
tic in  20'»  19'-9.  By  so  much  shorter,  then,  is  the  periodical  return  of 
our  seasons  than  the  true  sidereal  revolution  of  the  earth  round  the  sun. 
As  the  latter  period,  or  sidereal  year,  is  equal  to  365*  6"  9"  9'-6,  it  fol. 
lows,  then,  that  the  former  must  be  only  365*  5"  48"  49' "7  j  and  this  is 
what  is  meant  by  the  tropical  year. 

(384.)  We  have  already  mentioned  that  the  longer  axis  of  the  ellipse 
described  by  the  earth  has  a  slow  motion  of  ll"-8  per  annum  in  advance. 
From  this  it  results,  that  when  the  earth,  setting  out  from  the  perihelion, 
has  completed  one  sidereal  period,  the  perihelion  will  have  moved 
forward  by  ll"-8,  which  arc  must  be  described  by  the  earth  before  it  can 
again  reach  the  perihelion.  In  so  doing,  it  occupies  4"  39" -7,  and  this 
must  therefore  be  added  to  the  sidereal  period,  to  give  the  interval  between 
two  consecutive  returns  to  the  perihelion.  This  interval,  then,  is  365'* 
6"  IS™  49''3,'  and  is  what  is  called  the  anomalistic  year.  All  these 
periods  have  their  uses  in  astronomy;  but  that  in  which  mankind  in 
general  are  most  interested  is  tlie  tropical  year,  on  which  the  return  of 
the  seasons  depends,  and  which  we  thus  perceive  to  be  a  compound  phe- 
nomenon, depending  chiefly  and  directly  on  the  annual  revolution  of  the 
earth  round  the  sun,  but  subordinately  also,  and  indirectly,  on  its  rotation 
round  its  own  axis,  which  is  what  occasions  the  precession  of  the  equi- 
noxes; thus  affording  an  instructive  example  of  the  way  in  which  a 
motion,  once  admitted  in  any  part  of  our  system,  may  be  traced  in  its 
influence  on  others  with  which  at  first  sight  it  could  not  possibly  be  sup- 
posed to  have  any  thing  to  do. 

(385.)  As  a  rough  consideration  of  the  appearance  of  the  earth  points 
out  the  general  roundness  of  its  form,  and  more  exact  inquiry  has  led  us 
first  to  the  discovery  of  its  elliptic  figure,  and,  in  the  further  progress  of 
refinement,  to  the  perception  of  minuter  local  deviations  from  thnt  figure ; 
so,  in  investigating  the  solar  motions,  the  first  notion  we  obtain  is  that  of 
an  orbit,  generally  speaking,  round,  and  not  far  from  a  circle,  which,  on 
more  careful  and  exact  examination,  proves  to  be  an  ellipse  of  small  excen- 
tricity,  and  described  in  conformity  with  certain  laws,  as  above  stated. 
Still  minuter  inquiry,  however,  detects  yet  smaller  deviations  again  from 
this  form  and  from  these  laws,  of  which  wo  have  a  specimen  in  the  slow 
motion  of  the  axis  of  the  orbit  spoken  of  in  art.  372 ;  and  which  are 

'  These  numbers,  as  well  as  all  the  other  numerical  data  of  our  system,  are  taken 
from  Mr.  Daily's  Astronomical  Tables  and  FormulBB,  unless  the  contrary  is  expressed. 


PHYSICAL  CONSTITUTION  OF  THE  SUN. 


203 


generally  comprehended  under  the  name  of  perturbations  and  secular  in- 
equalities. Of  these  deviations,  and  their  causes,  we  shall  speak  here- 
after at  length.  It  is  the  triumph  of  physical  astronomy  to  have  rendered 
a  complete  account  of  them  all,  and  to  have  left  nothing  unexplained, 
either  in  the  motions  of  the  sun  or  in  those  of  any  other  of  the  bodies  of 
our  system.  But  the  nature  of  this  explanation  cannot  be  understooi)  till 
we  have  developed  the  law  of  gravitation,  and  carried  it  into  its  more 
direct  consequences.  This  will  be  the  object  of  our  three  following  chap- 
ters; in  which  we  shall  take  advantage  of  the  proximity  of  the  moon, 
and  its  immediate  connection  with  and  dependence  on  the  earth,  to  render 
it,  as  it  were,  a  stepping-stone  to  the  general  explanation  of  the  planet- 
ary movements.  We  shall  conclude  this  by  describing  what  is  known  of 
the  physical  constitution  of  the  sun. 

(386.)  When  viewed  through  powerful  telescopes,  provided  with 
coloured  glasses,  to  take  oflF  the  heat,  which  would  otherwise  injure  our 
eyes,  the  sun  is  observed  to  have  frequently  large  and  perfectly  black 
spots  upon  it,  surrounded  with  a  kind  of  border,  less  completely  dark, 
called  a  penumbra.  Some  of  these  are  represented  at  a,  h,  c,  d,  in  Plate 
I.  fig.  2.,  at  the  end  of  this  volume.  They  are,  however,  not  permanent. 
When  watched  from  day  to  day,  or  even  from  hour  to  hour,  they  appear 
to  enlarge  or  contract,  to  change  their  forms,  and  at  length  to  disappear 
altogether,  or  to  break  out  anew  in  parts  of  the  surface  where  none  were 
before.  In  such  cases  of  disappearance,  the  central  dark  spot  always  con- 
tracts into  a  point,  and  vanishes  before  the  border.  Occasionally  they 
break  up,  or  divide  into  two  or  more,  and  in  those  cases  offer  every  evi- 
dence of  that  extreme  mobility  which  belongs  only  to  the  fluid  state,  and 
of  that  excessively  violent  agitation  which  seems  only  compatible  with  the 
atmospheric  or  gaseous  state  of  matter.  The  scale  on  which  their  move- 
ments take  place  is  immense.  A  single  second  of  angular  measure,  as 
seen  from  the  earth,  corresponds  on  the  sun's  disc  to  461  miles ;  and  a 
circle  of  this  diameter  (containing  therefore  nearly  167000  square  miles) 
is  the  least  space  which  can  be  distinctly  discerned  or  'he  sun  as  a  visible 
area.  Spots  have  been  observed,  however,  whose  linear  diameter  has 
been  upwards  of  45000  miles;'  and  even,  if  some  records  arc  to  be 
trusted,  of  very  much  greater  extent,  That  such  a  spot  should  close  up 
in  six  weeks'  time  (for  they  seldom  last  much  longer),  its  borders  must 
approach  at  the  rate  of  more  than  1000  miles  a  day. 

(387.)  Many  other  circumstances  tend  to  corroborate  this  view  of  the 
subject.     The  part  of  the  sun's  disc  not  occupied  by  spots  is  far  from 


'  Mayer,  Obs.  Mar.  15,  1758. 
eter  =  25  diam.  solis. 


"  Ingens  macula  in  sole  conspiciebatur,  cujus  diam 


1 4i-  4  '11 


204 


OUTLINES  OP  ASTRONOMY. 


uniformly  bright.  Its  i/round  is  finely  mottled  with  an  appearance  of 
minute,  dark  dots,  or  pores,  which,  when  attentively  watched,  are  found 
to  be  in  a  constant  state  of  change.  There  is  nothing  which  represents 
so  faithfully  this  appearance  as  the  slow  subsidence  of  some  flocculent 
chemical  precipitates  in  a  transparent  fluid,  when  viewed  perpendicularly 
from  above :  so  faithfully,  indeed,  that  it  is  hardly  possible  not  to  be  im- 
pressed with  the  idea  of  a  luminous  medium  intermixed,  but  not  con- 
founded, with  a  transparent  and  non-luminous  atmosphere,  either  floating 
as  clouds  in  our  air,  or  pervading  it  in  vast  sheets  and  columns  like  flame, 
or  the  streamers  of  our  northern  lights,  directed  in  lines  perpendicular  to 
the  surface.  .: .  .^'^  ■■.:..■.     ,   ^    ^^^  .*  .: 

(388.)  Lastly,  in  the  neighbourhood  of  great  spots,  or  extensive  groups 
of  them,  largo  spaces  of  the  surface  are  often  observed  to  be  covered  with 
strongly  marked  curved  or  branching  streaks,  more  luminous  than  the 
rest,  called  facalsc,  and  among  these,  if  not  already  existing,  spots  fi'e< 
quently  break  out.  They  may,  perhaps,  be  regarded  with  most  proba- 
bility, as  the  ridges  of  immense  waves  in  the  luminous  regions  of  the 
eun's  atmosphere,  indicative  of  violent  agitation  in  their  neighbourhood. 
They  are  most  commonly,  and  best  seen,  towards  the  borders  of  the 
visible  disc,  and  their  appearance  is  as  represented  in  Plate  I.  fig.  1. 

(389.)  But  what  are  the  spots?  Many  fanciful  notions  have  been 
broached  on  this  subject,  but  only  one  seems  to  have  any  degree  of 
physical  probability,  viz.  that  they  are  the  dark,  or  at  least  comparatively 
dark,  solid  body  of  the  sun  itself,  laid  bare  to  our  view  by  those  immense 
fluctuations  in  the  luminous  regions  of  its  atmosphere,  to  which  it  appears 
to  be  subject.  Respecting  the  manner  in  which  this  disclosure  takes 
place,  different  ideas  again  have  been  advocated.  Lalande  (art.  3240) 
suggests,  that  eminences  in  the  nature  of  mountains  are  actually  laid 
bare,  and  project  above  the  luminous  ocean,  appearing  black  above  it, 
while  their  shoaling  declivities  produce  the  penumbrad,  where  the  lumi- 
nous fluid  is  less  deep.  A  fatal  objection  to  this  theory  is  the  uniform 
shade  of  the  penumbra  and  its  sharp  termination,  both  inwards,  where  it 
joins  the  spot,  and  outwards,  where  it  borders  on  the  bright  surface.  A 
more  probable  view  has  been  taken  by  Sir  William  Herschel,"  who  con- 
siders the  lumino\is  strata  of  the  atmosphere  to  be  sustained  far  above 
the  level  of  the  solid  body  by  a  transparent  elastic  medium,  carrying  on 
its  upper  surface  (or,  rather,  to  avoid  the  former  objection,  at  some  coU' 
siderahly  lower  level  within  its  depth)  a  cloudy  stratum  which,  being 
strongly  illuminated  from  above,  reflects  a  considerable  portion  of  the 
light  to  our  eyes,  and  forms  a  penumbra,  while  the  solid  body  shaded  by 
'■  » Phil.  Trans.  1801.  !/V 


V"  'P' 


NATURE   OF  THE  SUN'S  SPOTS. 


206 


Fig.  55. 


,  vv>v 


?^^■■^v|>vi^^;,^;,l-,^■■■■;;■ -v;'    ! 


the  clouds,  reflects  none.  (See  fff.)  The  temporary  removal  of  both 
the  strata,  but  more  of  the  upper  than  the  lower,  he  supposes  effected  by 
powerful  upward  currents  of  the  atmosphere,  arising,  perhaps,  from 
spiracles  in  the  body,  or  from  local  agitations. 

(390.)  When  the  spots  are  attentively  watched,  their  situation  on  the 
disc  of  the  sun  is  observed  to  change.  They  advance  regularly  towards 
its  western  limb  or  border,  where  they  disappear,  and  are  replaced  by 
others  which  tenter  at  the  eastern  limb,  and  which,  pursuing  their  respec- 
tive courses,  in  their  turn  disappear  at  the  western.  The  apparent 
rapidity  of  this  movement  is  not  uniform,  as  it  would  bo  were  the  spots 
dark  bodies  passing,  by  an  independent  motion  of  their  own,  between  the 
earth  and  the  sun ;  but  is  swiftest  in  the  middle  of  their  paths  across 
the  disc,  and  very  slow  at  its  borders.  This  is  precisely  what  would  be 
the  case  supposing  them  to  appertain  to  and  make  part  of  the  visible 
surface  of  the  sun's  globe,  and  to  be  carried  round  by  a  uniform  rotation 
of  that  globe  on  its  axis,  so  that  each  spot  should  describe  a  circle  parr^llel 
to  the  sun's  equator,  rendered  elliptic  by  the  effect  of  perspective.  Tiieir 
apparent  paths  also  across  the  disc  conform  to  this  view  of  their  nature, 
being,  generally  speaking,  ellipses,  much  elongated,  concentric  with  the 
sun's  disc,  each  having  one  of  its  chords  for  its  longer  axis,  and  all  these 
axes  parallel  to  each  other.  At  two  periods  of  the  year  only  do  the 
spots  appear  to  describe  straight  lines,  viz.  on  and  near  to  the  11th  of 
June  and  the  12th  of  December,  on  which  days,  therefore,  the  plane  of 
the  circle,  which  a  spot  situated  on  the  sun's  equator  describes  (and  con- 
sequently,  the  planft  of  that  equator  itself,)  passes  through  the  earth. 
Hence  it  is  obvious,  that  the  plane  of  the  sun's  equator  is  inclined  t?  that 
of  the  ecliptic,  and  intersects  it  in  a  line  which  passes  through  the  place 
of  the  earth  on  these  days.     The  situation  of  this  line,  or  the  line  of  the 


206 


OUTLINES   OF  ASTRONOMY. 


node$  of  the  sutCs  equator  as  it  is  called,  is,  therefore,  defined  by  the 
longitudes  of  the  earth  as  scon  from  the  sun  at  those  epochs,  which  are 
rospectivoly  80"  21'  and  260°  21'  (=80°  21' +  180°)  being,  of  course, 
diametrically  opposite  in  direction. 

(891.)  The  inclination  of  the  sun's  axis  (that  of  the  plane  of  its 
equator)  to  the  ecliptic  is  determined  by  ascertaining  the  proportion  of 
the  longer  and  the  shorter  diameter  of  the  apparent  ellipse,  described  by 
any  remarkable,  well-defined  spot;  in  order  to  do  which,  its  apparent 
place  on  the  sun's  disc  must  be  very  precisely  ascertained  by  micrometric 
measures,  repeated  from  day  to  day  as  long  as  it  continues  visible  (usually 
about  12  or  13  days,  according  to  the  magnitude  of  the  spots,  which 
always  vanish  by  the  effect  of  foreshortening  before  they  attain  the  actual 
border  of  the  disc — but  the  larger  spots  being  traceable  closer  to  the  limb 
than  the  smaller.')  The  reduction  of  such  observations,  or  the  conclu- 
sion from  them  of  the  element  in  question,  is  complicated  with  the  effect 
of  the  earth's  motion  in  the  interval  of  the  observations,  and  with  its 
situation  in  the  ecliptic,  with  respect  to  the  line  of  nodes.  For  simplicitj^ 
we  will  suppose  the  earth  situated  as  it  is  on  the  10th  of  March,  in  a  line 
at  right  angles  to  that  of  the  nodes,  i.  e.  in  the  heUocentrio  longitude 
170°  21',  and  to  remain  there  stationary  during  the  whole  passage  of  a 
spot  across  the  dUo.    In  this  case  the  axis  pf  rotation  of  the  sun  will  be 


T\g.  66. 


V 


situated  in  a  plane  passing  through  the  earth  and  ,9t  right  angles  to  the 
plane  of  the  ecliptic.  Suppose  C  to  represent  the  sun's  centre,  P  C  p 
its  asds,  E  0  the  line  of  sight,  P  N  Q  A^  S  a  cection  of  the  sun  passing 

'  The  great  spot  of  December,  1719,  is  stated  to  have  been  seen  as  a  notch  in  the 
limb  of  thesur» 


-•Pf 


OF  THE  SUM  S  ROTATION  ON  ITS  AXIS. 


20: 


through  the  earth,  and  Q  a  spot  situated  on  ita  equator,  and  in  that  pkne 
and  ooDsequently  in  the  middle  of  its  apparent  path  across  the  disc.  It 
the  axis  of  rotation  were  perpendicular  to  the  ecliptic,  as  N  S,  this  spot 
would  be  at  A,  and  would  be  seen  projected  on  C,  the  centre  of  the  sun. 
It  is  actually  at  Q,  projected  upon  D,  at  an  apparent  distance  0  D  to  the 
north  of  the  centre,  which  is  the  apparent  smaller  semi-axis  of  the  ellipse 
described  by  the  epot,  which  being  known  by  miorometrio  measurementi 

C  D 

the  value  of  t^-^  or  the  cosine  of  Q  0  N,  the  inclination  of  the  sun's 

equator  becomes  known,  C  N  being  the  apparent  semi-diameter  of  the 
sun  at  the  time.  At  this  epoch,  moreover,  the  northern  half  of  the  circle 
described  by  the  spot  is  visible  (the  southern  passing  behind  the  body  of 
the  sun,)  and  the  south  pole  p  of  the  sun  is  within  the  visible  hemi- 
sphere. This  is  the  case  in  the  whole  interval  from  December  11th  to 
July  12th,  during  which,  the  visual  ray  falls  upon  the  southern  side  of 
the  sun's  equator.  The  contrary  happens  in  the  other  half  year,  from 
July  12th  to  December  11th,  and  this  is  what  is  understood  when  we  say 
that  the  ascending  node  (denoted  Q)  of  the  sun's  equator  lies  in  80°  21' 
longitude — a  spot  on  the  equator  passing  that  node  being  then  in  the  act 
of  ascending  from  the  southern  to  the  northern  side  of  the  plane  of  the 
ecliptic — such  being  the  conventional  language  of  astronomers  in  speaking 
of  these  matters. 

(392.)  If  the  observations  are  made  at  other  seasons  (which,  however, 
are  the  less  favourable  for  this  purpose  the  more  remote  they  are  from 
the  epochs  here  assigned) ',  when,  moreover,  as  in  strictness  is  necessary, 
the  motion  of  the  e^rth  in  the  interval  of  the  measures  is  allowed  for  (as 
for  a  change  of  the  point  of  sight) ;  the  calculations  requisite  to  deduce 
the  situation  of  the  axis  in  space,  and  the  duration  of  the  revolution 
around  it,  become  much  more  intricate,  and  it  would  be  beyond  the  scope 
of  this  work  to  enter  into  them.*  According  to  the  best  determinations 
we  possess,  the  inclination  of  the  sun's  equator  to  the  ecliptic  is  about  7° 
20'  (its  nodes  being  as  above  stated),  and  the  period  of  rotation  25  days 
7  hours  48  minutes.' 

(893.)  The  region  of  the  spots  is  confined,  generally  speaking,  withm 

about  25°  on  either  side  of  the  sun's  equator;  beyond  80°  they  are  very 

* . "'-      ■.■.'''■  ■         ''  ''" 

'  See  the  theory  in  Lelnnd's  Astronomy,  art.  3258,  and  the  formuln  of  computation 
in  a  paper  by  Petersen  Schumacher's  Nachrichten,  No.  419. 

'Bianchi  (Schumacher's  Nach.  483),  agreeing  with  Laugier.    Leiambre  makes  it 
25*  Oi>  17" ;  Petersen,  25''  4'*  30".    The  incUnation  of  the  axis  is  uncertain  to  half  a 
degree,  and  the  node  to  several  degrees.    The  continual  changes  in  the  spots  them 
selves  cause  this  uncertainty. 


208 


OUTLINES   OF  A8TR0N0MT. 


rarely  scon;  in  the  polar  rngionn,  Dover.  Tbo  actual  equator  of  the  snn 
is  also  lcH8  fre(|ucntly  vibitcd  by  spots  than  the  udjacent  zones  on  cither 
Hide,  and  a  very  material  difTurcneo  in  their  frequency  and  luagnitado 
subsists  in  its  northern  and  southern  hemisphere,  Ihoqe  on  thn  nortliora 
preponderating  in  both  respects.  The  zone  comprised  between  ttio  11th 
and  15th  degree  to  the  northward  of  the  equator  is  particularly  fertile  in 
large  and  durable  spots.  These  circumstances,  as  well  as  the  frequent 
occurronce  of  a  more  or  less  regular  arrangement  of  the  spots,  when 
numerous,  in  the  manner  of  belts  parallel  to  the  equator,  point  evidently 
to  phyHical  peculiarities  in  certain  parts  of  the  sun's  body  morb  favourable 
than  in  others  to  the  production  of  the  spots,  on  the  one  hand ;  and  on 
the  other,  to  a  general  influence  of  its  rotation  on  its  axis  as  a  determining 
cause  of  their  distribution  and  arrangement,  and  would  appear  indicative 
of  a  8ystem  of  movements  in  the  fluids  which  constitute  its  lumint/us 
surface  bearing  no  remote  analogy  to  our  trade  winds  —  from  whatever 
cause  arising.     (See  art.  239.  et  scq.)  ^ 

(394.)  The  duration  of  individual  spots  is  commonly  not  grtiit;  some 
are  formed  and  disappear  within  the  limit  of  a  single  transit  across  the 
disc — but  such  are  for  themost  part  small  and  insignificant.  Frequently 
they  make  one  or  two  revolutions,  being  recognized  at  their  reappearance 
by  their  situation  with  respect  to  the  equator,  their  configurations  inter  sr, 
their  size,  or  other  peculiarities,  as  well  as  by  the  interval  elapsing  be- 
tween their  disappearance  at  ono  J'".o  and  reapp'"aranco  on  tbo  other.  In 
a  few  rare  cases,  however,  tluy  have  been  watched  round  many  revolu- 
tions. The  great  spot  of  177*J  appeared  during  six  months,  and  one  and 
the  same  ijioiip  was  obsorved  in  1840  by  Schwabe  to  return  eight 
times.'  It  has  been  surmised,  with  considerable  apparent  probability,  that 
some  spots,  at  least,  arc  generated  again  and  again,  at  distant  intervals  of 
time,  over  the  same  identical  points  of  the  sun's  body  (as  hurricanes,  for 
;xamplc,  are  known  to  affect  given  localities  on  the  earth's  surface,  and  to 
pursue  definite  tracks).  The  uncertainty  which  still  prevails  with  respect 
to  the  exact  duration  of  its  rotation  renders  it  very  diflRmlt  to  obtain  con- 
vincing evidence  of  this;  nor,  ind'^od,  can  it  be  cxpectf^d,  itr^iil  bv  bring- 
ing together  into  one  connected  view  the  recorded  st  of  *;:iC  iun's  sur- 
face during  a  very  long  period  of  time,  and  comparing  together  remarka- 
ble bj^ots  which  have  appeared  on  {he  same  paraUd,  some  precise  periodic 
tijie  Esball  be  found  which  shall  exactly  conciliate  numerous  and  well- 
chara'itjriztd  appearances.  The  inquiry  is  one  of  singular  interest,  as 
there  can  ;>e  n'^  reasonable  doubt  that  the  supply  of  light  and  heat 

•B(,bun,  J.'ficu.  No.  418,  ;  150.  The  recent  papers  of  Biela,  Capocci,  Schwabe, 
Pasicrli',  and  S>hmidt,  in  th^t  collection,  will  be  found  highly  interesting. 


OF  THB  sum's  spots. 


209 


aiFordcd  to  our  globe  stands  in  intimate  connexion  with  tb'>M  processei 
n  1  are  taking  place  on  the  solitr  sui  fitce,  and  to  which  the  spots  in 
some  way  or  other  owe  their  origin 

(896.)  Above  the  luminous  Hiirfaco  of  ')\c  'un,  and  be  region  in  which 
(he  spots  reside,  there  are  strong  lu  lioationa  ot  the  fxir'tionoe  of  a  gaseous 
atmosphere  having  a  somewhat  imperfect  transparency.  When  the  whole 
disc  of  the  sun  is  seen  at  once  through  a  telescope  magnifying  moduratoly 
ODOUgh  to  allow  it,  and  with  a  darkening  glass  such  s  to  suITt  it  to  be 
oontetnp1^te(^  with  perfect  comfort,  it  is  very  evident  ,  'lat  the  I  orders  of 
\H  Ji  ^  if-o  much  less  luminous  than  the  centre.  That  'us  is  no  illusion 
it,  bhot  n  Ig'  projecting  the  sun's  image  undarkened  and  moderately  mag- 
Tiifiod,  so  13  to  occupy  a  circle  two  or  three  inches  in  diameter,  on  a  nhoct 
ui  tvbite  paper,  taking  care  to  have  it  well  in  focus,  when  tl  >  samu  ap- 
pearance will  be  observed.  This  can  only  arise  from  the  circ  .mfercntial 
rays  having  undergone  the  absorptive  action  of  a  much  greater  thickness 
ot  Bome  imperfectly  transparent  envelope  (due  to  greater  obli«<iity  of 
their  passage  through  it)  than  the  central.  —  But  a  still  more  con  inoing 
aud  indeed  decisive  evidence  is  offered  by  the  phsenomena  attchiing  a 
total  eclipse  of  the  sun.  Such  eclipses  (as  will  be  shown  hereafte  )  are 
produced  by  the  interposition  of  the  dark  body  of  the  moon  betweci  the 
earth  and  sun,  the  moon  being  large  enough  to  cover  and  surpass,  y  a 
very  small  breadth,  the  whole  disc  of  the  sun.  Now  when  this  t;  cea 
place,  were  there  no  vaporous  atmosphere  capable  of  reflecting  any  11^  ht 
about  the  sun,  the  sky  ought  to  appear  totally  dark,  since  (as  will  here- 
after abundantly  appear)  there  is  not  the  smallest  reason  for  believing  tl  e 
moon  to  have  any  atmosphere  capable  of  doing  so.  So  far,  however,  s 
this  from  being  the  case,  that  a  bright  ring  or  corona  of  light  is  seen, 
fading  gradually  away,  as  represented  in  PI.  I.  fig.  8.,  which  (in  casea 
where  the  moon  is  not  centrally  superposed  on  the  sun)  is  observed  to  be 
concentric  with  the  latter,  not  the  former  body.  This  corona  was  beauti- 
fully seen  in  the  eclipse  of  July  7,  1842,  and  with  this  most  remarkable 
addition  —  witnessed  by  every  spectator  in  Pavia,  Milan,  Vienna,  and 
tisewhere :  there  distinct  and  very  conspicuous  rose-coloured  protuberances 
(as  represented  in  the  figure  cited)  were  seen  to  project  beyond  the  dark 
limb  of  the  moon,  likened  by  some  to  flames,  by  others  to  mountains,  but 
which  their  enormous  magnitude  (for  to  have  been  seen  at  all  by  the 
naked  eye  tl.tiir  height  must  have  exceeded  40,000  miles),  and  their  faint 
degree  of  illumination,  olearly  prove  to  have  been  cloudy  masses  of  th^ 
most  excesswe  tenuity,  and  which  doubtless  owed  their  support,  and  proba< 
bly  their  exiistenco.  to  suou  an  atmosphere  as  we  are  now  speaking  of. 

(396.)  That  the  teuiperaturo  at  the  visible  surface  of  the  sun  cannoi 
14 


u\ 


210 


OUTLINES   OF  ASTRONOMY. 


be  uiherwjse  tban  very  elevated,  much  more  so  than  any  artificial  heat 
produced  in  our  furnaces,  or  by  chemical  or  galvanic  processes,  we  have 
indications  of  several  distinct  kinds :  1st,  From  the  law  of  decrease  of 
radiant  heat  and  light,  which,  being  inversely  as  the  squares  of  the  dis- 
tances, it  follows,  that  the  heat  received  on  a  given  area  exposed  at  the 
distance  ot  the  earth,  and  on  an  equal  area  at  the  visible  surface  of  the 
sun,  must  be  in  the  proportion  of  the  area  of  the  sky  occupied  by  the 
sun's  apparent  disc  to  the  whole  hemisphere,  or  as  1  to  about  800000. 
A  far  less  intensity  of  solar  radiation,  collected  in  the  focus  of  a  burning 
glass,  suffices  to  dissipate  gold  and  platina  in  vapour.  2dly,  From  the 
facility  with  which  the  calorific  rays  of  the  sun  traverse  glass,  a  property 
which  is  found  to  belong  to  the  heat  of  artificial  fires  in  the  direct  pro- 
portion of  their  intensity.'  3dly,  From  the  fact,  that  the  most  vivid 
flames  disappear,  and  the  most  intensely  ignited  solids  appear  only  as 
black  spots  on  the  disc  of  the  sun  when  held  between  it  and  the  eye.^ 
From  the  last  remark  it  follows,  that  the  body  of  the  sun,  however  dark 
it  may  appear  when  seen  through  its  spots,  may,  nevertheless,  be  in  \ 
state  of  most  intense  ignition.  It  does  not,  however,  follow  of  necessity 
that  it  mmt  be  so.  The  contrary  is  at  least  physically  possible.  A.  per- 
fectly reflective  canopy  would  efiectually  defend  it  from  the  radiation  of 
the  luminous  regions  above  its  atmosphere,  and  no  heat  would  be  con- 
ducted downwards  through  a  gaseous  medium  increasing  rapidly  in 
density.  That  the  penumbral  clouds  are  highly  reflective,  the  fact  of 
their  visibility  in  such  a  situation  can  leave  no  doubt. 

(897.)  As  the  magnitude  of  the  sun  has  been  measured,  and  (as  we 
shall  hereafter  see)  its  weight,  or  quantity  of  ponderable  matter,  ascer- 
tained, so  also  attempts  have  been  made,  and  not  wholly  without  success, 
from  the  heat  actually  communicated  by  its  rays  to  given  surfaces  of 
material  bodies  exposed  to  their  vertical  action  on  the  earth's  surface,  to 
estimate  the  total  expenditure  of  heat  by  that  luminary  in  a  given  time. 
The  result  of  such  experiments  has  been  thus  announced.  Supposing  a 
cylinder  of  ice  45  miles  in  diameter,  to  be  continually  darted  into  the  sun 
u-ith  the  velocity  of  light,  and  that  the  water  produced  by  its  fusion  were 


'  By  direct  measurement  with  the  actinometer,  I  find  that  out  of  1000  calorific  solar 
rays,  816  penetrate  a  sheet  of  plate  glass  0'12  inch  ihick ;  and  that  of  1000  rays  which 
have  passed  through  one  such  plate,  859  are  capable  of  passing  through  another.  H. 
1827. 

*  The  ball  of  ignited  quicklime,  in  Lieutenant  Drummond's  oxy-hydrogen  lamp, 
gives  the  nearest  imitation  of  the  solar  splendour  which  has  yet  been  produced.  The 
appearance  of  this  against  the  sun,  was,  however,  as  described  in  an  imperfect  trial  at 
which  I  was  present.  The  experiment  ought  to  be  repeated  under  favourable  circum 
■tanccs.— iVivfe  to  the  ed,  of  1833 


TERRESTRIAL  EFFECTS   OF  THE   SUN'S   RADIATION.         211 

continually  carried  off,  the  heat  now  given  off  constantly  by  radiation 
would  then  be  wholly  expended  in  its  liquefaction,  on  the  one  hand,  so  as 
to  leave  no  radiant  surplus ;  while,  on  th«  other,  the  actual  temperature 
at  its  surface  would  undergo  no  diminution. 

(398.)  This  immense  escape  of  heat  by  radiation,  we  may  remark,  will 
fully  explain  the  constant  state  of  tumultuous  agitation  in  which  the  fluids 
composing  the  visible  surface  are  maintained,  and  the  continual  generation 
and  filling  in  of  the  pores,  without  having  recourse  to  internal  causes. 
The  mode  of  action  here  alluded  to  is  perfectly  represented  to  the  eye  in 
the  disturbed  subsidence  of  a  precipitate,  as  described  in  art.  887,  when 
the  fluid  from  which  it  subsides  is  warm,  and  losing  heat  from  its  surface. 
(399.)  The  sun's  rays  are  the  ultimate  source  of  almost  every  motion 
which  takes  place  on  the  surface  of  the  earth.  By  its  heat  are  produced 
all  winds,  and  those  disturbances  in  the  electric  equilibrium  of  the  atmo- 
sphere which  give  rise  to  the  phenomena  of  lightning,  and  probably  also 
to  those  of  terrestrial  magnetism  and  the  aurora.  By  their  vivifying 
action  vegetables  are  enabled  to  draw  support  from  inorganic  matter,  and 
become,  in  their  turn  the  support  of  animals  and  of  man,  and  the  sources 
of  those  great  deposits  of  dynamical  efficiency  which  are  laid  up  for 
human  use  in  our  coal  strata.'  By  them  the  waters  of  the  sea  are  made 
to  circulate  in  vapour  through  the  air,  and  irrigate  the  land,  producing 
springs  and  rivers.  By  them  are  produced  all  disturbances  of  the 
chemical  equilibrium  of  the  elements  of  nature,  which,  by  a  series  of 
compositions  and  decompositions,  give  rise  to  new  products,  and  originate 
a  transfer  of  materials.  Even  the  slow  degradation  of  the  solid  con- 
stituents of  the  surface,  in  which  its  chief  geological  changes  consist,  is 
almost  entirely  due  on  the  one  hand  to  the  abrasion  of  wind  and  rain,  and 
the  alternation  of  heat  and  frost ;  on  the  other  to  the  continual  beating 
of  the  sea  waves,  agitated  by  winds,  the  results  of  solar  radiation.  Tidal 
action  (itself  partly  due  to  the  sun's  agency)  exercises  here  a  compara- 
tively slight  influence.  The  effect  of  oceanic  currents  (mainly  originating 
in  that  influence,)  though  slight  in  abrasion,  is  powerful  in  diffusing  and 
transporting  the  matter  abraded;  and  when  we  consider  the  immense 
transfer  of  matter  so  produced,  the  increase  of  pressure  over  large  spaces 
in  the  bed  of  the  ocean,  and  diminution  over  corresponding  portions  of 
the  land,  we  are  not  at  a  loss  to  perceive  how  the  elastic  power  of  sub- 
terranean fires,  thus  repressed  on  the  one  hand  and  relieved  on  the  other, 
may  break  forth  in  points  where  the  resistance  is  barely  adequate  to  their 
retention,  and  thus  bring  the  phenomena  of  even  volcanic  activity  under 
the  general  law  of  solar  influence.* 

'  So  in  the  edition  of  1833.  *  So  in  the  edition  of  1833. 


I     I 


1  ,*v* 


212 


OUTLINES  OF  ASTRONOi'^. 


(400.)  The  great  mystery,  however,  is  to  conceive  how  so  enomiJtis  a 
conflagration  (if  such  it  be)  can  be  kept  up.  Every  discovery  in  chemi- 
cal science  here  leaves  us  completely  at  a  loss,  or  rather,  seems  to  remove 
farther  the  prospect  of  probable  explanation.  If  conjecture  might  be 
hazarded,  we  should  look  rather  to  the  known  possibility  of  an  indefinite 
generation  of  heat  by  friction,  or  to  its  excitement  by  the  electric  dis- 
charge, than  to  any  actual  combustion  of  ponderable  fuel,  whether  solid 
or  gaseous,  for  the  origin  of  the  solar  radiation.* 


'  Electricity  traversing  excessively  rarefied  air  or  vapours,  gives  out  light,  and, 
doubtless,  also  heat.  May  not  a  continual  current  of  electric  matter  be  constantly 
circulating  in  the  sun's  immediate  neighbourhood,  or  traversing  the  planetary  spaces, 
and  exciting,  in  the  upper  regions  of  its  atmosphere,  those  phenomena  of  which,  on 
however  diminutive  a  scale,  we  have  yet  an  unequivocal  manifestation  in  our  aurora 
borealis.  The  possible  analogy  of  the  solar  light  to  that  of  the  aurora  has  been 
distinctly  insisted  on  by  the  late  Sir  W.  Herschel,  in  his  paper  already  cited.  It  would 
be  a  highly  curious  subject  of  experimental  inquiry,  how  far  a  mere  reduplication  of 
sheets  of  dame,  at  a  distance  one  behind  the  other  (by  which  their  light  might  he 
brought  to  any  required  intensity,)  would  communicate  to  the  heat  of  the  resulting 
compound  ray  the  penetrating  character  which  distinguishes  the  solar  caloriiic  rnya. 
We  may  also  observe,  that  the  tranquillity  of  the  sun's  polar,  as  compared  with  its 
equatortal  regions  (if  its  spots  be  really  atmospheric,)  cannot  be  accounted  for  by  its 
rotation  on  its  axis  only,  but  must  arise  from  some  cause  external  to  the  luminous  sur- 
face of  the  sun,  as  we  see  the  belts  of  Jupiter  and  Saturn,  and  our  trade-winds  arise 
from  a  cause,  external  to  these  planets,  combining  itself  with  their  rotation,  which 
alone  can  produce  no  motions  when  once  the  form  of  equilibrium  is  attained. 

The  prismatic  analysis  of  the  solar  beam  exhibits  in  the  spectrum  a  series  of  "  fixed 
Knes,"  totally  unUke  those  which  belong  to  the  light  of  any  known  terrestrial  flame. 
This  may  hereafter  lead  us  to  a  clearer  insight  into  its  origin.  But,  before  we  can 
draw  any  conclusions  from  such  an  indication,  we  must  recollect  that  previous  to 
reaching  us  it  has  undergone  the  whole  absorptive  action  of  our  atmosphere,  as  well 
as  of  the  sun's.  Of  the  latter  we  know  nothing,  and  may  conjecture  every  thing ; 
but  of  the  blue  colour  of  the  former  we  are  sure ;  and  if  this  be  an  inherent  (t.  e.  an 
absorptive)  colour,  the  air  must  be  expected  to  act  on  the  spectrum  after  the  analogy 
of  other  coloured  media,  which  often  (and  especially  light  blue  media)  leave  unab- 
sorbed  portions  separated  by  dark  intervals.  It  deserves  inquiry,  therefore,  whether 
some  or  all  the  fixed  lines  observed  by  WoUaston  and  Fraunhofer  may  not  have  their 
origin  in  our  own  atmosphere.  Experiments  made  on  lofty  mountains,  or  the  cars  of  bal- 
loons, on  the  one  hand,  and  on  the  other  with  reflected  beams  which  have  been  made 
to  traverse  several  miles  of  additional  air  near  the  surface,  would  decide  this  point. 
The  absorptive  effect  of  the  sun's  atmosphere,  and  possibly  also  of  the  medium  sur- 
rounding it  (whatever  it  be)  which  resists  the  motions  of  comets,  cannot  be  thus 
eliminated. — Note  to  the  edition  of  1833. 


or   THE   MOON. 


213 


CHAPTER  Vn. 

OP  THE   MOON. — ITS   SIDEREAL  PERIOD. — ITS  APPARENT  DIAMETER. 

—  ITS  PARALLAX,  DISTANCE,  AND  REAL  DIAMETER. —  FIRST  AP- 
PROXIMATION TO  ITS  ORBIT.  —  AN  ELLIPSE  ABOUT  THE  EARTH  IN 
THE  FOCUS. — ITS  EXCENTRICITY  AND  INCLINATION.  —  MOTION  OP 
ITS  NODES  AND  APSIDES. — OP  OCCULTATIONS  AND  SOLAR  ECLIPSES 
GENERALLY.  —  LIMITS  WITHIN  WHICH  THEY  ARE  POSSIBLE.  —  THEY 
PROVE  THE  MOON  TO  BE  AN  OPAKE  SOLID. — ITS  LIGHT  DERIVED 
FROM  THE  SUN.  —  ITS  PHASES.  —  SYNODIC  REVOLUTION  OR  LUNAR 
MONTH. — OF  ECLIPSES  MORE  PARTICULARLY. — THEIR  PHENOMENA. 

—  THEIR  PERIODICAL  RECURRENCE.  —  PHYSICAL  CONSTITUTION  OP 
THE  MOON. — ITS  MOUNTAINS  AND  OTHER  SUPERFICIAL  FEATURES. 
— INDICATIONS  OF  FORMER  VOLCANIC  ACTIVITY. — ITS  ATMOSPHERE. 

—  CLIMATE.  —  RADIATION  OP  HEAT  FROM  ITS  SURFACE. — ROTATION 
ON  ITS  OWN  AXIS.  —  LIBRATION. — APPEARANCE  OP  THE  EARTH 
FROM  IT. 

(401.)  The  moon,  like  the  sun,  appears  to  advance  among  the  stars 
\Tith  a  movement  contrary  to  the  general  diurnal  motion  of  the  heavens, 
but  much  more  rapid,  so  as  to  be  very  readily  perceived  (as  we  have 
before  observed)  by  a  few  hours'  cursory  attention  on  any  moonlight 
night.  By  this  continual  advance,  which,  though  sometimes  quicker, 
Bometimes  slower,  is  never  intermitted  or  reversed,  it  makes  the  tour  of 
the  heavens  in  a  mean  or  average  period  of  27*  T"*  43"  ll»-5,  returning, 
in  that  time,  to  a  position  among  the  stars  nearly  coincident  with  that  it 
bad  before,  and  which  would  be  exactly  so,  but  for  reasons  presently  to 
be  stated. 

(402.)  The  moon,  then,  like  the  sun,  apparently  describes  an  orbit 
round  the  earth,  and  this  orbit  cannot  be  very  different  from  a  circle,  be- 
cause the  apparent  angular  diameter  of  the  full  moon  is  not  liable  to  any 
great  extent  of  variation. 

(40C.)  The  distance  of  the  moon  from  the  earth  is  concluded  from  its 
horizontal  parallax,  which  may  be  found  either  directly,  by  observations 
at  remote  geographical  stations,  exactly  similar  to  those  described  "n  art. 
855,  in  the  case  of  the  sun,  or  by  means  of  the  phaenomena  called  occul- 


iii-iiffl 


la'   ' 


0  5*    jfet* 


214 


OUTLINES   OF  •ASTRONOMY. 


tations,  from  which  also  its  apparent  diameter  is  most  readily  and  cor- 
rectly found.  From  such  observations  it  results  that  the  mean  or  average 
distance  of  the  centre  of  the  moon  from  that  of  the  earth  is  59-9643  of 
the  earth's  equatorial  radii,  or  about  237,000  miles.  This  distance 
great  as  it  is,  is  little  more  than  one-fourth  of  the  diameter  of  the  sun's 
body,  so  that  the  globe  of  the  sun  would  nearly  twice  include  the  whole 
orbit  of  the  moon;  a  consideration  wonderifuUy  calculated  to  raise  our 
ideas  of  that  stupendous  luminary !  ,       j 

(404.)  The  distance  of  the  moon's  centre  from  an  observer  at  any 
station  on  the  earth's  surface,  compared  with  its  apparent  angular  diameter 
as  measured  from  that  station,  will  give  its  real  or  linear  diameter.  Now, 
the  former  distance  is  easily  calculated  when  the  distance  from  the  earth's 
centre  is  known,  and  the  apparent  zenith  distance  of  the  moon  also  deter- 
mined by  observation  j  for  if  we  turn  to  the  figure  of  art.  339,  and  suppose 
S  the  moon,  A  the  station,  and  C  the  earth's  centre,  the  distance  S  C,  and 
the  earth's  radius  C  A,  two  sides  of  the  triangle  A  C  S  are  given,  and  the 
angle  CAS,  which  is  the  supplement  of  Z  A  S,  the  observed  zenith  dis- 
tance, whence  it  is  easy  to  find  A  S,  the  moon's  distance  from  A.  From 
such  observations  and  calculations  it  results,  that  the  real  diameter  of  tbe 
moon  is  2160  miles,  or  about  0-2729  of  that  of  the  earth,  whence  it  follows 
that,  the  bulk  of  the  latter  being  considered  as  1,  that  of  the  former  will 
be  00204,  or  about  ^^g.  The  difference  of  the  apparent  diameter  of  the 
moon,  as  seen  from  the  earth's  centre  and  from  any  point  of  its  surface, 
is  technically  called  the  augmentation  of  the  apparent  diameter,  and  its 
maximum  occurs  when  the  moon  is  in  the  zenith  of  the  spectator.  Her 
mean  angular  diameter,  as  seen  from  the  centre,  is  81'  7",  and  is  always 
s=  0-545  X  her  horizontal  parallax. 

(405.)  By  a  series  of  observations,  such  as  described  in  art.  403,  if 
continued  during  one  or  more  revolutions  of  the  moon,  its  real  distance 
may  be  ascertained  at  every  point  of  its  orbit ;  and  if  at  the  same  time  its 
apparent  places  in  the  heavens  be  observed,  and  reduced  by  means  of  its 
parallax  to  the  earth's  centre,  their  angular  intervals  will  become  known, 
so  that  the  path  of  the  moon  may  then  be  laid  down  on  a  chart  supposed 
to  represent  the  plane  in  which  its  orbit  lies,  just  as  was  explained  in  the 
case  of  the  solar  ellipse  (art.  349.)  Now,  when  this  is  done,  it  is  found 
that,  neglecting  certain  small  (though  very  perceptible)  deviations  (of 
which  a  satisfactory  account  will  hereafter  be  rendered),  the  form  of  the 
apparent  orbit,  like  that  of  the  sun,  is  elliptic,  but  considerably  more 
eccentric,  the  eccentricity  amounting  to  0-05484  of  \\iq  mean  distance,  or 
the  major  semi-axis  of  the  ellipse,  and  the  earth's  centre  being  situated  in 
its  focus. 


OP  THE  MOON  S  MOTION. 


215 


(406.)  The  plane  in  which  this  orbit  lies  is  not  the  ecliptic,  however, 
but  is  inclined  to  it  at  an  angle  of  5°  8'  48",  which  is  called  the  incli« 
nation  of  the  lunar  orbit,  and  intersects  it  in  two  opposite  points,  which 
are  called  its  nodes— the  ascending  node  being  that  in  which  the  moon 
passes  from  the  southern  side  of  the  ecliptic  to  the  northern,  and  the 
descending  the  reverse.  The  points  of  the  orbit  at  which  the  moon  is 
nearest  to,  and  farthest  from,  the  earth,  are  called  respectively  its  perigee 
and  apogee,  and  the  line  joining  them  and  the  earth  of  the  line  of  apsides. 

(407.)  There  are,  however,  several  remarkable  circumstances  which 
interrupt  the  closeness  of  the  analogy,  which  cannot  fail  to  strike  the 
reader,  between  the  motion  of  the  moon  around  the  earth,  and  of  the 
earth  around  the  sun.  In  the  latter  case,  the  ellipse  described  remains, 
daring  a  great  many  revolutions,  unaltered  in  its  position  and  dimensions ; 
or,  at  least,  the  changes  which  it  undergoes  are  not  perceptible  but  in  a 
course  of  very  nice  observations,  which  have  disclosed,  it  is  true,  the 
existence  of  "  perturbations,"  but  of  so  minute  an  order,  that,  in  ordinary 
parlance,  and  for  common  purposes,  we  may  leave  them  unconsidered. 
But  this  cannot  be  done  in  the  case  of  the  moon.  Even  in  a  single  revo« 
lution,  its  deviation  from  a  perfect  ellipse  is  very  sensible.  It  does  not 
return  to  the  same  exact  position  among  the  stars  from  which  it  set  out, 
thereby  indicating  a  continual  change  in  the  plane  of  its  orbit.  And,  in 
effect,  if  we  trace  by  observation,  from  month  to  month,  the  point  where 
it  traverses  the  ecliptic,  we  shall  find  that  the  nod^s  of  its  orbit  are  in  a 
continual  state  of  retreat  upon  the  ecliptic.  Suppose  0  to  be  the  earth, 
and  Ab  ad  that  portion  of  the  plane  of  the  ecliptic  which  is  intersected 
by  the  moon,  in  its  alternate  passages  through  it,  from  south  to  north,  and 
vice  versd;  and  let  A  B  C  D  E  F  be  a  portion  of  the  moon's  orbit,  em- 


E    • 


tiJ* 


bracing  a  complete  sidereal  revolution.  Suppose  it  to  set  out  from  the 
ascending  node,  A  j  then,  if  the  orbit  lay  all  in  one  plane,  passing  through 
0,  it  would  have  a,  the  opposite  point  in  the  ecliptic,  for  its  descending 
node ;  after  passing  which,  it  would  again  ascend  at  A.  But,  in  fact,  its 
real  path  carries  it  not  to  a,  but  along  a  certain  curve,  A  B  C,  to  C,  A 


216 


OUTLINES   OF  A6TR0N0MT. 


^1 


I 


I 

'i 

'■■1 

IS 


point  ia  the  ecliptic  less  than  180°  distant  from  A;  so  that  the  angle 
A  0  C,  or  the  arc  of  longitude  described  between  the  ascending  and  the 
descending  node,  is  somewhat  less  than  180°.  It  then  pursues  its  course 
below  the  ecliptic,  along  the  curve  C  D  E,  and  rises  again  above  it,  not  at 
the  point  c,  diametrically  opposite  to  C,  but  at  a  point  E,  less  advanced  in 
longitude.  On  the  whole,  then,  the  arc  described  in  longitude  betreen 
two  consecutive  passages  from  south  to  north,  through  the  plane  of  the 
ecliptic,  fulls  short  of  860°  by  the  angle  A  0  E ;  or,  in  other  words,  the 
ascending  node  appears  to  have  retreated  in  one  lunation  on  the  plane  of 
the  ecliptic  by  that  amount.  To  complete  a  sidereal  revolution,  then,  it 
must  still  go  on  to  describe  an  arc,  E  F,  on  its  orbit,  which  will  no  longer, 
however,  bring  it  exactly  back  to  A,  but  to  a  point  somewhat  above  it,  or 
having  north  latitude. 

(408.)  The  actual  amount  of  this  retreat  of  the  moon's  node  is  about 
3'  10"-64  per  diem,  on  an  average,  and  in  a  period  of  6793-39  mean 
solar  days,  or  about  18*6  years,  the  ascending  node  is  carried  round  in  a 
direction  contrary  to  the  moon's  motion  in  its  orbit  (or  from  east  to  welst) 
over  a  whole  circumference  of  the  ecliptic.  Of  course,  in  the  middle  of 
this  period  the  position  of  the  orbit  must  have  been  precisely  reversed 
from  what  it  was  at  the  beginning.  Its  apparent  path,  then,  will  Ho 
among  totally  different  stars  and  constellations  at  different  parts  of  this 
period ;  anJ  this  kind  of  spiral  revolution  being  continually  kept  up,  it 
will,  at  one  time  or  other,  cover  with  its  disc  every  point  of  the  heavens 
within  that  limit  of  latitude  or  distance  from  the  ecliptic  which  its  inclina- 
tion permits;  that  is  to  say,  a  belt  or  zone  of  the  heavens,  of  10°  18'  in 
breadth,  having  the  ecliptic  for  its  middle  line.  Nevertheless,  ii  still 
remains  truo  that  the  actual  plac^  of  the  moon,  in  consequence  of  this 
motion,  deviates  in  a  single  revolution  very  little  from  what  it  would  be 
were  the  nodes  at  rest.  Supposing  the  moon  to  set  out  from  its  node  A, 
its  latitude,  when  it  comes  to  F,  having  completed  a  revolution  in  longi- 
tude, will  not  exceed  8';  which,  though  small  in  a  single  revolution, 
accumulates  in  its  effect  in  a  succession  of  maay  :  it  is  to  account  for,  and 
represent  geometrically,  this  deviation,  that  the  motion  of  the  nodes  is 
devised. 

(409.)  The  moon's  orbit,  then,  is  not,  strictly  speaking,  an  ellipse 
returning  into  itself,  by  reason  of  the  variation  of  the  plane  in  which  it 
lies,  and  the  motion  of  its  nodes.  But  even  laying  aside  this  considera- 
tion, the  axis  of  the  ellipse  ia  itself  constantly  changing  its  direction  in 
space,  as  has  already  been  stated  of  the  solar  ellipse,  but  much  more 
rapidly ;  making  a  complete  revolution,  in  the  same  direction  with  the 
moon's  own  motion,  in  3232-5753  mean  solar  days,  or  about  nine  years, 


MOTION  OF  THE  NODES  AND  APSIDES. 


217 


being  about  8°  of  angular  motion  in  a  whole  revolution  of  the  moon. 
This  is  a  phenomenon  known  by  the  name  of  the  revolution  of  the  moon's 
apsides.  Its  cause  will  be  hereafter  explained.  Its  immediate  effect  is 
to  produce  a  variation  in  the  moon's  distance  from  the  earth,  which  is  not 
included  in  the  laws  of  exact  elliptio  motion.  In  a  single  revolution  of 
the  moon,  this  variation  of  distance  is  trifling ;  but  in  the  course  of  many 
it  becomes  considerable,  as  is  easily  seen,  if  we  consider  that  in  four  years 
and  a  half  the  position  of  the  axis  will  be  completely  reversed,  and  the 
apogee  of  the  moon  will  occur  where  the  perigee  occurred  before. 

(410.)  The  best  way  to  form  a  distinct  conception  of  the  moon's  motion 
is  to  regard  it  as  describing  an  ellipse  about  the  earth  in  the  focus,  and, 
at  the  same  time,  to  regard  this  ellipse  itself  to  be  in  a  twofold  state  of 
revolution ,  1st,  in  its  own  plane,  by  a  continual  advance  of  its  axis  in 
that  plane ;  and  2dly,  by  a  continual  tilting  motion  of  the  plane  itself, 
exactly  similar  to,  but  much  more  rapid  than,  that  of  the  earth's  equator 
produced  by  the  conical  motion  of  its  axis  described  in  art.  317. 

(411.)  As  the  moon  is  at  a  very  moderate  distance  from  us  (astronomi- 
cally speaking),  and  is  in  fact  our  nearest  neighbour,  while  the  sun  and 
stars  are  in  comparison  immensely  beyond  it,  it  must  of  necessity  happen, 
that  at  one  time  or  other  it  must  pass  over  and  occult  or  eclipse  every  star 
and  planet  within  the  zone  above  described  (and,  as  seen  from  the  surface 
of  the  earth,  even  somewhat  beyond  it,  by  reason  of  parallax,  which  may 
throw  it  apparently  nearly  a  degree  either  way  from  its  place  as  seen  from 
the  centre,  according  to  the  observer's  station).  Nor  is  the  sun  itself 
exempt  from  being  thus  hidden,  whenever  any  part  of  the  moon's  disc, 
in  this  her  tortuous  course,  comes  to  overlap  any  part  of  the  space  occu- 
pied in  the  heavens  by  that  luminary.  On  these  occasions  is  exhibited 
the  most  striking  and  impressive  of  all  the  occasional  phenomena  of 
astronomy,  an  eclipse  of  the  sun,  in  which  a  greater  or  less  portion,  or 
even  in  some  rare  conjunctures  the  whole,  of  its  disc  is  obscurcid,  and,  as  it 
were,  obliterated,  by  the  superposition  of  that  of  the  moon,  which  appears 
upon  it  as  a  circularly-terminated  black  spot,  producing  a  temporary  dimi- 
nution of  daylight,  or  even  nocturnal  darkness,  so  that  the  stars  appear  as 
if  at  midnight.  In  other  cases,  when,  at  the  moment  that  the  moon  is 
centrally  superposed  on  the  sun,  it  so  happens  that  her  distance  from 
the  earth  is  such  as  to  render  her  angular  diameter  less  than  the  sun's, 
the  very  singular  phenomenon  of  an  annular  solan'  eclipse  takes  place, 
when  the  edge  of  the  sun  appears  for  a  few  minutes  as  a  narrow  ring  of 
light,  projecting  on  all  sides  beyond  the  dark  circle  occupied  by  the  moon 
in  its  centre. 

(412.)  A  solar  eclipse  can  only  happen  when  the  sun  and  moon  are  m 


• 


i-  ! 


218 


OUTLINES   OP  ASTRONOMY. 


I!  I 


i 


conjunction,  that  is  to  say,  have  the  same,  or  nearly  the  same,  position  in 
the  heavens,  or  the  same  longitude.  It  appears  by  art.  409  that  this 
condition  can  only  be  fulfilled  at  the  time  of  a  new  moon,  though  it  by  no 
means  follows,  that  at  ever^/  conjunction  there  must  be  an  eclipse  of  the 
sun.  If  the  lunar  orbit  coincided  with  the  ecliptic,  this  would  be  the 
cose,  but  as  it  is  inclined  to  it  at  an  angle  of  upwards  of  5°,  it  is  evident 
that  the  conjunction,  or  equality,  of  longitudes,  may  take  place  when  the 
moon  is  in  the  part  of  her  orbit  too  remote  from  the  ecliptic  to  permit  the 
discs  to  meet  and  overlap.  It  is  easy,  however,  to  as':ign  the  limits 
within  which  an  eclipse  is  possible.  To  this  end  we  must  consider,  that, 
by  the  effect  of  parallax,  the  moon's  apparent  edge  may  be  thrown  ii 
any  direction,  according  to  a  spectator's  geographical  station,  by  any 
amount  not  exceeding  the  horizontal  parallax.  Now,  this  comes  to  the 
same  (so  far  as  the  possibility  of  an  eclipse  is  concerned)  as  if  the  ap- 
parent diameter  of  the  moon,  seen  from  the  earth's  centre,  were  dilated 
by  twice  its  horizontal  parallax ;  for  if,  when  so  dilated,  it  can  touch  or 
overlap  tha  sun,  there  must  be  an  eclipse  at  some  part  or  other  of  the 
earth's  surface.  If,  then,  at  the  moment  of  the  nearest  conjunction,  the 
geocentric  distance  of  the  centres  of  the  two  luminaries  do  not  exceed  the 
sum  of  their  semidiameters  and  of  the  moon's  horizontal  parallax,  there 
will  be  an  eclipse.  This  sum  is,  at  its  maximum,  about  1°  34'  27".  In 
the  spherical  triangle  S  N  M,  then,  in  which  S  is  the  sun's  centre,  M  the 
moon's,  S  N  the  ecliptic,  M  N  the  moon's  orbit,  and  N  the  node,  we  may 

Fig.  58. 


suppose  the  angle  N  S  M  a  right  angle,  S  M  =  1°  34'  27",  and  the  angle 
M  N  S  =  5°  8'  48",  the  jaclination  of  the  orbit.  Hence  we  calculate 
S  N,  which  conies  cat  16"  58'.  If,  then,  at  the  moment  of  the  new 
moon,  the  moon's  node  is  farther  from  the  sun  in  longitude  than  this 
limit.,  there  can  be  no  eclipse ;  if  within,  there  may,  and  probably  will,  at 
some  part  or  other  'of  the  earth.  To  ascertain  precisely  whether  there 
will  or  not,  and,  if  there  be,  how  great  will  be  the  part  eclipsed,  the  solar 
and  lunar  tables  must  be  consulted,  the  place  of  the  node  and  the  semi- 
diameters  exactly  ascertained,  and  the  local  parallax,  and  apparent  aug- 
mentation of  the  moon's  diameter  due  to  the  difference  of  her  distance 


OOOULTATION  OF  A  STAR  BT  THB  MOON. 


219 


from  the  observer  and  from  the  centro  of  the  earth  (which  may  s  '>unt 
to  a  sixtieth  part  of  her  horizontal  diameter),  determined ;  after  wii.  .a  it 
is  easy,  from  the  above  considerations,  to  calculate  the  amount  overlapped 
of  the  two  discs,  and  their  moment  of  contact. 

(413.)  The  calculation  of  the  occultation  of  a  star  depends  on  similar 
considerations.  An  occultation  is  possible,  when  the  moon's  course,  as 
seen  from  the  earth's  centre,  carries  her  within  a  distance  from  the  star 
equal  to  the  sum  of  her  semidiameter  and  horizontal  parallax ,  and  it  will 
hippen  at  any  particular  spot,  when  her  apparent  path,  as  seen  from  that 
spot,  carries  her  centre  within  a  distance  equal  to  the  sum  of  her  aug- 
mented semidiameter  and  actual  parallax.  The  details  of  these  calcula- 
tions, which  are  somewhat  troublesome,  must  be  sought  elsewhere.' 

(414.)  The  phenomenon  of  a  solar  eclipse  and  of  an  occultation  are 
highly  interesting  and  instructive  in  a  physical  point  of  view.  They 
teach  us  that  the  moon  is  an  opaque  body,  terminated  by  a  real  and  sharply 
defined  surface  intercepting  light  like  a  solid.  They  prove  to  us,  also, 
that  at  those  t  33  when  We  cannot  see  the  moon,  she  really  exists,  and 
pursues  her  course,  and  that  when  we  see  her  only  as  a  crescent,  however 
narrow,  the  whole  globular  body  is  there,  filling  up  the  deficient  outline, 
though  unseen.  For  occultAtions  take  place  indifferently  at  the  dark  and 
bright,  the  visible  aud  invisible  outline,  whichever  happens  to  be  towards 
the  direction  in  which  the  moon  is  moving )  with  this  only  difference,  that 
a  star  occulted  by  the  bright  limb,  if  the  phenomenon  be  watched  with  a 
telescope,  gives  notice,  by  its  gradual  approach  to  the  visible  edge,  when 
to  expect  its  disappearance,  while,  if  occulted  at  the  dark  limb,  if  the 
moon,  at  least,  be  more  than  a  few  days  old,  it  is,  as  it  were,  extinguished 
in  mid-air,  without  notice  or  visible  cause  for  its  disappearance,  which,  as 
it  happens  instantaneously,  and  without  the  slightest  previous  diminution 
of  its  light,  is  always  surprising ;  and,  if  the  star  be  a  large  and  bright 
one,  even  startling  from  its  suddenness.  The  re-appearance  of  the  star, 
too,  when  the  moon  has  passed  over  it,  takes  place  in  those  cases  when 
the  bright  side  of  the  moon  is  foremost,  not  at  the  concave  outline  of  the 
crescent,  but  at  the  invisible  outline  of  the  complete  circle,  and  is  scarcely 
less  surprising,  from  its  suddenness,  than  its  disappearance  in  the  other 
case. 

'  Woodhouse's  Astronomy,  vol.  i.    See  also  Trans.  Ast.  Soc.  vol.  i.  p.  325. 

'  There  is  an  optical  illusion  of  a  very  strange  and  unaccountable  nature  which  has 
often  been  remarked  in  occultations.  The  star  appears  to  advance  actually  upon  and 
within  the  edge  of  the  disc  before  it  disappears,  and  that  sometimes  to  a  considerable 
depth.  I  have  never  myself  witnessed  this  singular  effect,  but  it  rests  on  most  une- 
quivocal testimony.  I  have  called  it  an  optical  illusion ;  but  it  is  barely  potsible  that  a 
star  may  shine  on  such  occasions  through  deep  fissures  in  the  substance  of  the  moon. 


220 


OUTLINES  OF  ASTRONOMT. 


k 


m 


(415.)  The  existenco  of  the  complete  circle  of  the  disc,  even  when 
the  moon  ia  not  full,  does  not,  however,  rest  only  on  the  evidence  of 
oooultations  and  eclipses.  It  may  be  seen,  when  the  moon  is  orescent  or 
waning,  a  few  days  before  and  after  the  new  moon,  with  the  naked  eye, 
as  a  pale  round  body,  to  which  the  orescent  seems  attached,  and  soue« 
what  projecting  beyond  its  outline  (which  is  an  optical  illusion  arising 
from  the  greater  intensity  of  its  light.)  The  cause  of  this  appearance 
will  presently  be  explained.  Meanwhile  the  fact  is  sufficient  to  show 
that  the  moon  is  not  inherently  luminoi^.s  like  the  sun,  but  that  her  light 
is  of  an  adventitious  nature.  And  its  crescent  form,  increasing  regularly 
from  a  narrow  semicircular  line  to  a  complete  circular  disc,  corresponds  to 
the  appearance  a  globe  would  present,  one  hemisphere  of  which  was 
black,  the  other  white,  when  differently  turned  towards  the  eye,  no  as  to 
piosout  a  greater  or  less  portion  of  each.  The  obvious  conclusion  from 
this  is,  that  the  moon  is  such  a  globe,  one  half  of  which  is  brightened  by 
the  rays  of  some  luminary  sufficiently  distant  to  enlighten  the  complete 
hemisphere,  and  sufficiently  intense  to  give  it  the  degree  of  ypiendour  we 
see.  Now,  the  sun  alone  is  competent  to  such  an  effect.  Its  distance 
and  light  suffice ;  and,  moreover,  it  is  invariably  observed  that,  when  a 
crescent,  the  bright  edge  is  towards  the  sun,  and  that  in  proportion  as 
the  moon  in  her  monthly  course  becomes  more  and  more  distant  from  the 
sun,  the  breadth  of  the  crescent  increases,  and  v'""  versd. 

(416.)  The  sun's  distance  being  28984  radii  of  the  earth,  and  the 
moon's  only  60,  the  former  is  nearly  400  times  the  latter.  Lines,  there- 
fore, drawn  from  the  sun  to  every  part  of  the  moon's  orbit  may  be 
regarded  as  very  nearly  parallel.'  Suppose,  now,  0  to  be  the  earth, 
A  B  C  D,  &c.  various  positions  of  the  moon  in  its  orbit,  and  S  the  sun, 
at  the  vast  distance  above  stated ;  as  is  shown,  then,  in  the  figure,  the 
hemisphere  of  the  lunar  globe  turned  towards  it  (on  the  right)  will  he 
bright,  the  opposite  dark,  wherever  it  may  stand  in  its  orbit.  Now,  in 
the  position  A,  when  in  conjunction  with  the  sun,  the  dark  part  is 
entirely  turned  towards  0,  and  the  bright  from  it.     In  this  case,  then, 


The  occultations  of  close  double  stars  ought  to  be  narrowly  watched,  to  see  whether 
both  individuals  are  thus  projected,  as  well  as  for  oth  ir  purposes  connected  with  their 
theory.  I  will  only  hint  at  one,  viz.  that  a  double  Uar,  too  clone  to  be  seen  divided 
with  any  telescope,  may  yet  be  detected  to  be  double  uy  the  mode  of  its  disapppear- 
ance.  Should  a  considerable  star,  for  instance,  instead  of  undergoing  instantaneous 
and  complete  extinction,  go  out  by  two  distinct  steps,  following  close  upon  each  other; 
first  losing  a  portion,  then  the  whole  remainder  of  its  light,  we  may  be  sure  it  is  a 
double  star,  though  we  cannot  see  the  individuals  separately. — Note  to  the  edit,  of  1833. 
'  The  angle  subtended  by  the  moon's  orbit,  as  seen  from  the  sun,  (in  the  mean  state 
of  things,)  is  only  17'  12". 


PHASES   OF  THE  MOON  EXPLAINED. 


221 


Fig.  59. 


the  moon  is  not  seen,  it  is  new  moon.  When  the  moon  has  come  to  0, 
half  the  bright  and  half  the  dark  hemisphere  are  presented  to  0,  and  the 
same  in  the  opposite  situation  G :  these  are  the  first  and  third  quarters 
of  the  moon.  Lastly,  when  at  E,  the  whole  bright  face  is  towards  the 
earth,  the  whole  dark  side  from  it,  and  it  is  then  seen  wholly  bright  or 
full  moon.  In  the  intermediate  positions  B  D  F  H,  the  portions  of  the 
bright  face  presented  to  0  will  be  at  first  less  than  half  the  visible  sur- 
face, then  greater,  and  finally  less  again,  till  it  vanishes  altogether,  as  it 
comes  round  again  to  A. 

(417.)  These  monthly  changes  of  appearance,  or  phases,  as  they  are 
called,  arise,  then,  from  the  moon,  an  opaque  body,  being  illuminated  on 
one  side  by  the  sun,  and  reflecting  from  it,  in  all  directions,  a  portion  of 
the  light  so  received.  Nor  let  it  be  thought  surprising  that  a  solid  sub- 
stance thus  illuminated  should  appear  to  shine  and  again  illuminate  the 
earth.  It  is  no  more  than  a  white  cloud  does  standing  off  upon  the  clear 
blue  sky.  By  day,  the  moon  can  hardly  be  distinguished  in  brightness 
from  such  a  cloud ;  and,  in  the  dusk  of  the  evening,  clouds  catching  the 
last  rays  of  the  sun  appear  with  a  dazzling  splendour,  not  inferior  to  the 
seeming  brightness  of  the  moon  at  night.*  That  the  earth  sends  also 
such  a  light  to  the  moon,  only  probably  more  powerful  by  reason  of  its 
greater  apparent  size',  is  agreeable  to  optical  principles,  and  explains  the 

'  The  actual  illuminatiun  of  the  lunar  surface  is  not  much  superior  to  that  of  weathered 
sandstone  rock  in  full  sunshine.  I  have  frequentlj^  compared  the  moon  setting  behind 
the  grey  perpendicular  fayade  of  the  Table  Mountain,  illuminoted  by  the  sim  just  risen 
in  the  opposite  quarter  of  the  horizon,  when  it  has  been  scarcely  distinguishable  iii 
brightness  from  the  rock  in  contact  with  it.  The  sun  and  moon  being  nearly  at  equal 
altitudes  and  the  atmosphere  perfectly  free  from  cloud  or  vapour,  its  effect  is  alike  on 
both  luminaries.  (H.  1848). 

*The  apparent  diameter  of  the  moon  is  32'  from  the  earth ;  that  of  the  earth  seen 
from  the  moon  is  twice  her  horizontal  parallax,  or  1"  54'.    The  apparent  surfaces 
therefore,  are  as  (114)':  (32)»,  or  aa  13  :  1  nearly. 


m 

•  if? 

•  1  # 
an 


222 


OUTLINES  OF  ASTRONOMT. 


appearance  of  the  dark  portion  of  the  young  or  waning  moon  completing 
its  crcscuut  (art.  413).  For,  when  the  moon  is  nearly  new  to  the  earth, 
the  latter  (so  to  speak)  is  nearly  full  to  the  former;  it  then  illuminates  ita 
dark  half  by  strong  earth-Uijht ;  and  it  is  a  portion  of  this,  reflected  back 
again,  which  makes  it  visible  to  us  in  the  twilight  sky.  As  the  mooa 
gains  age,  the  earth  offers  it  a  less  portion  of  its  bright  side,  and  the  phe- 
nomenon in  question  dies  away. 

(418.)  The  lunar  month  is  determined  by  the  recurrence  of  its  phases: 
it  reckons  from  new  moon  to  new  moon ;  that  is,  from  leaving  its  conjunc- 
tion  with  the  sun  to  its  return  to  conjunction.  If  the  sun  stood  still,  like 
a  fixed  star,  the  interval  between  two  conjunctions  would  bo  the  same  aa 
the  period  of  the  moon's  sidereal  revolution  (art.  401) ;  but,  as  the  sun 
apparently  advances  in  the  heavens  in  the  same  direction  with  the  moon, 
only  slower,  the  latter  has  more  than  a  complete  sidereal  period  to  perform 
to  con\o  up  with  the  sun  again,  and  will  require  for  it  a  longer  time,  which 
is  the  lunar  month,  or,  as  it  is  generally  termed  in  astronomy,  a  »i/nodical 
period.  The  difference  is  easily  calculated  by  considering  that  the  super- 
fluous arc  (whatever  it  be)  is  described  by  the  sun  with  the  velocity  of 
0°*985C5  per  diem,  in  the  same  time  that  the  moon  describes  that  arc 
plus  a  complete  revolution,  with  her  velocity  of  13°*17640per  diem;  and, 
the  times  of  description  being  iuentical,  the  spaces  ore  to  each  other  in  the 
proportion  of  the  velocities.  Let  V  and  v  bo  the  mean  angular  velocities, 
X  the  superfluous  arc ;  then  V  :  v  : :  1  +  a; : a;;  and  V  —  i; :  v  : :  1  :  x, 


X 


whence  x  is  found,  and  —  =  the  time  of  describing  a;,  or  the  difference  of 

the  sidereal  and  synodical  periods.  From  these  data  a  slight  knowledge 
of  arithmetic  will  suffice  to  derive  the  arc  in  question,  and  the  time  of  its 
description  by  the  moon ;  which  being  the  excess  of  the  synodic  over  the 
sidereal  period,  the  former  will  be  had,  and  will  appear  to  be  29''  12" 
44-  2-87. 

(419.)  Supposing  the  position  of  the  nodes  of  the  moon's  orbit  to 
permit  it,  when  the  moon  stands  at  A  (or  at  the  new  moon),  it  will  inter- 
cept a  part  or  the  whole  of  the  sun's  rays,  and  cause  a  solar  eclipse.  Oa 
the  other  hand,  when  at  E  (or  at  the  full  moon),  the  earth  0  will  inter- 
cept the  rays  of  the  sun,  and  cast  a  shadow  on  the  moon,  thereby  causing 
a  lunar  eclipse.  And  this  is  perfectly  consonant  to  fact,  such  eclipses 
never  happening  but  at  the  exact  time  of  the  full  moon.  But,  what  is 
still  more  remarkable,  as  confirmatory  of  the  position  of  the  earth's  sphe- 
ricity, this  shadow,  which  we  plainly  see  to  enter  upon  and,  as  it  were, 
eat  away  the  disc  of  tb'»  moon,  is  always  terminated  by  a  circular  outline, 
though,  from  the  greater  size  of  the  circle,  it  is  only  partially  seen  at  any 


SOLAR   AND    LUNAR   ECLIP8E§^ 


228 


one  time.     Now,  a  body  «  )yi«h  ulwajN  oasts  a  circular  shadow  niii^t  itself 
be  gphorioal. 

(4'20.)  Eclipses  of  the  sun  are  best  understood  by  regarding  (h«  sun 
and  moon  as  two  independent  luniinarios,  each  moving  according  to  known 
laws,  and  viewed  from  the  oarth :  but  it  is  also  instructive  to  consider 
eclipses  general!)-  as  arising  from  the  shadow  of  one  body  thrown  on  ano- 
ther by  a  luminary  much  larger  than  either.  Suppose  then,  A  B  to 
represent  the  sun,  and  C  D  a  s|iLc'rical  body,  whether  earth  or  moon,  illu- 
minated by  it.  If  we  join  and  prolong  AC,  BDj  since  AB  is  greater 
than  C  D,  thcso  lines  will  meet  in  a  point  E,  more  or  less  distant  from 
the  body  0  D,  according  to  its  size,  and  within  the  space  C  E  D  (which 
represents  a  cone,  since  C  D  and  A  B  are  spheres),  there  will  be  a  total 
shadow.  This  shadow  is  called  the  umbra,  and  a  spectator  situated  within 
it  can  see  no  part  of  the  sun's  disc.    Beyond  the  umbra  are  two  diverging 

Fig.  60. 


spaces  (or  rather,  a  portion  of  a  single  conical  space,  having  K  for  its 

vertex),  where  if  a  spectator  be  situated,  as  at  M,  he  will  see  a  portion 

only  (A  0  N  P)  of  the  sun's  surface,  the  rest  (B  0  N  P)  being  obscured 

by  the  earth.     He  will,  therefore,  receive  only  partial  sunshine;  and  the 

more,  the  nearer  he  is  to  the  exterior  borders  of  that  cone  which  is  called 

i\iQ penumbra.     Beyond  this  he  will  see  the  whole  sun,  and  be  in  full 

illumination.     All  these  circumstances  may  be  perfectly  well  shown  by 

holding  a  small  globe  up  in  the  sun,  and  receiving  its  shadow  at  diflferent 

distances  on  a  sheet  of  paper. 

(421.)  In  a  lunar  eclipse  (represented  in  the  upper  figure),  the  moon 

is  seen  to  enter'  iha pcnumhra  first,  and,  by  degrees,  get  involved  in  the 

umbra,  the  former  bordering  the  latter  like  a  smoky  haze.   At  this  period 

of  the  eclipse,  and  while  yet  a  considerable  part  of  the  moon  remains 

'  The  actual  contact  with  the  penumbra  is  never  seen  ;  the  defalcation  of  hght  comes 
on  so  V6.y  gradually  that  it  is  not  till  when  already  deeply  immersed,  that  it  is  perceived 
to  be  senvbly  darkened* 


\¥.  ^ 


f;* 


J  '  f 


;  hit 


224 


OUTLINES  OF  ASTRONOMY. 


UDobscured,  the  portion  involved  in  the  umbra  is  invisible  to  the  naked 
eye,  though  still  perceptible  in  a  telescope,  and  of  a  dark  grey  hue.  But 
as  the  eclipse  advances,  and  the  enlightened  part  diminishes  in  extent,  and 
grows  gradually  more  and  more  obscured  by  the  advance  of  the  penumbra^ 
the  eye,  relieved  from  its  glare,  becomes  more  sensible  to  feeble  impres- 
sions of  light  and  colour ;  and  phenomena  of  a  remarkable  and  instruc- 
tive character  begin  to  be  developed.  The  umbra  is  seen  to  be  very  far 
from  totally  dark :  and  in  its  faint  illumination  it  exhibits  a  gradation  of 
colour,  being  bluish,^  or  even  (by  contrast)  somewhat  greenish,  towards 
the  borders  for  a  space  of  about  4'  or  5'  of  apparent  angular  breadth 
inwards,  thence  passing,  by  delicate  but  rapid  gradation,  through  rose  red 
to  a  fiery  or  copper-coloured  glow,  like  that  of  dull  red-hot  iron.  As 
the  eclipse  proceeds  this  glow  spreads  over  the  whole  surface  of  the  moon, 
which  then  becomes  on  some  occasions  so  strongly  illuminated,  as  to  cast 
a  very  sensible  shadow,  and  allow  the  spots  on  its  surface  to  be  perfectly 
well  distinguished  through  a  telescope. 

(422.)  The  cause  of  these  singular,  and  sometimes  very  beautiful 
appearances,  is  the  refraction  of  the  sun's  light  in  passing  through  our 
atmosphere,  which  at  the  same  time  becomes  coloured  with  the  hues  of 
sunset  by  the  absorption  of  more  or  less  of  the  violet  and  blue  rays,  as  it 
passes  through  strata  nearer  or  more  remote  from  the  earth's  surface,  and 
therefore,  more  or  less  loaded  with  vapour.  To  show  this,  let  A  D  a  be 
a  section  of  the  cone  of  the  umbra,  and  F  B  h/  of  the  penumbra,  through 
their  common  axis  D  E  S,  passing  through  the  centres  E  S  of  the  earth 
and  sun,  and  let  K  M  A;  be  a  section  of  these  cones  at  a  distance  E  M  from 
E,  equal  to  the  radius  of  the  moon's  orbit,  or  60  radii  of  the  earth.' 
Taking  this  radius  for  unity,  since  E  S,  the  distance  of  the  sun,  is  23984, 
and  the  semidiameter  of  the  sun  111^  such  units,  we  readily  calculate 
D  E=218,  D  M=158,  for  the  distances  at  which  the  apex  of  the  geome- 
ti-ical  umbra  lies  behind  the  earth  and  the  moon  respectively.  We  also 
find  for  the  measure  of  the  angle  E  D  B,  15'  46",  and  therefore  D  B  E= 
89°  44'  14",  whence  also  we  get  M  C  (the  linear  semidiameter  of  the 
^^w^ira)=0•725  (or  in  miles  2864),  and  the  angle  OEM,  its  apparent 
angular  semidiameter  as  seen  from  E=41'  30".  And  instituting  similar 
calculations  for  the  geometrical  penumbra  we  get  M  K= 1-005  (3970 
miles),  and  K  E  M  57'  36";  and  it  may  be  well  to  remember  that  the 
doubles  of  these  angles,  or  the  mean  angular  diameters  of  the  umbra  and 
penumbra,  are  described  by  the  moon  with  its  mean  velocity  in  2"  43", 
and  S**  17"  respectively,  which  are  therefore  the  respective  durations  of 

'  Ti.e  figure  is  unavoidably  drawn  out  of  all  proportion,  and  the  angles  violently 
«xaggerated.    The  reader  auould  endeavour  to  draw  the  figure  in  its  true  proporUoni. 


PHENOMENA  OF  A  LUNAR  ECLIPSE. 
Fig.  61. 


225 


the  total  and  partial  obscuration  of  any  one  point  of  the  moon's  disc  in 
traversing  centrally  the  geometrical  shadow. 

(423.)  Were  the  earth  devoid  of  atmosphere,  the  whole  of  the  phe- 
nomena of  a  lunar  eclipse  would  consist  in  these  partial  or  total  obscura- 
tions. Within  the  space  C  c  the  whole  of  the  light,  and  within  K  C  and 
c  h  a  greater  or  less  portion  of  it,  would  be  intercepted  by  the  solid  body 
B6  of  the  earth.  The  refracting  atmosphere,  however,  extends  from 
B,  h,  to  a  certain  unknown,  but  very  small  distance  B  H,  6  ^,  which,  acting 
as  a  convex  lens,  of  gradually  (and  very  rapidly)  decreasing  density,  dis- 
perses all  that  comparatively  small  portion  of  light  which  falls  upon  it 
over  a  space  bounded  externally  by  H  </,  parallel  and  very  nearly  coinci- 
dent with  B  F,  and  internally  by  a  line  B  z,  the  former  representing  the 
extreme  exterior  ray  from  the  limb  a  of  the  sun,  the  latter,  the  extreme 
interior  ray  from  the  limb  A.  To  avoid  complication,  however,  we  will 
trace  only  the  courses  of  rays  which  just  graze  the  surface  at  B,  viz:  Bz 
from  the  upper  border,  A,  and  B  v  from  the  lower,  a,  of  the  sun.  Each 
of  these  rays  is  bent  inwards  from  its  original  course  by  double  the 
amount  of  the  horizontal  refraction  (33')  i.  e.  by  1°  6',  because,  in 
passing  from  B  out  of  the  atmosphere,  it  undergoes  a  deviation  equal  to 
that  at  entering,  and  in  the  same  direction.  Instead,  therefore,  of  pur- 
suing the  courses  B  D,  B  F,  these  rays  respectively  will  occupy  the  posi- 
tions B  z  y,  B  V,  making,  with  the  aforesaid  lines,  the  angles  D  B  J,  F  B  v, 
each  1^  G'.  Now  we  have  found  DBE=  89^  44'  14"  and  therefore 
15 


\ 


r 


I 


( '1 


226 


OUTLINES   OF  ASTRONOMY. 


FBE(  =  DBJ+  angular  diam.  of  o)  =  90''  17'  17",  consequently 
the  angles  E  By  and  E  B  v  will  be  respectively  88°  38'  14"  and  89°  U' 
17"  from  which  we  conclude  E^;  =  4203  and  Ev  =  88-89,  the  former 
fulling  short  of  the  moon's  orbit  by  I'i  07,  and  the  latter  surpassing  it  by 
28-89  radii  of  the  earth. 

(424.)  The  penumbra,  therefore,  of  rays  refracted  at  B,  will  be  spread 
over  the  space  v  B  y,  that  at  H  over  g  H  d,  and  at  the  intermediate 
points,  over  similar  intermediate  spaces,  and  through  this  compound  of 
superposed  penumbrso  the  moon  passes  during  the  whole  of  its  path 
through  the  geometrical  shadow,  never  attaining  the  absolute  umbra 
B  a  6  at  all.  Without  going  into  detail  as  to  the  intensity  of  tbe 
refracted  rays,  it  is  evident  that  the  totality  of  light  so  thrown  into  the 
shadow  is  to  that  which  the  earth  intercepts,  as  the  area  of  a  circular 
section  of  the  atmosphere  to  that  of  a  diametrical  section  of  the  earth 
itself,  and,  therefore,  at  all  events  but  feeble.  And  it  is  still  further 
enfeebled  by  actual  clouds  suspended  in  that  portion  of  the  air  which 
forms  the  visible  border  of  the  earth's  disc  as  seen  from  the  moon,  as 
well  as  by  the  general  want  of  transparency  caused  by  invisible  vapour, 
which  is  especially  effective  in  the  lowermost  strata,  within  three  or  four 
miles  of  the  surface,  and  which  will  impart  to  all  the  rays  they  transmit, 
the  ruddy  hue  of  sunset,  only  of  double  the  depth  of  tint  which  wo 
admire  in  our  glowing  sunsets,  by  reason  of  the  rays  having  to  traverse 
twice  as  great  a  thickness  of  atmosphere.  This  redness  will  be  most 
intense  at  the  points  a*,  y,  of  the  moon's  path  through  the  umbra,  and 
will  thence  degrade  very  rapidly  outwardly,  over  the  spaces  x  c,y  Q,  less 
so  inwardly,  over  x  y.  And  at  C,  c,  its  hue  will  be  mingled  with  the 
bluish  or  greenish  light  which  the  atmosphere  scatters  by  irregular  dis- 
persion, or  in  other  words  by  our  tidlltjht  (art.  44).  Nor  will  the  phe- 
nomenon be  uniformly  conspicuous  at  all  times.  Supposing  a  generally 
and  deeply  clouded  state  of  the  atmosphere  around  the  edge  of  the  earth's 
disc  visible  from  the  moon  {i.  e.  around  thf.t  great  circle  of  the  earth,  in 
which,  at  the  moment  the  sun  is  in  the  horizon,)  little  or  no  refracted 
light  may  reach  the  moon.'  Supposing  that  circle  partly  clouded  aud 
partly  clear,  patches  of  red  light  corresponding  to  the  clear  portions  will 
be  thrown  into  the  umbra,  and  may  give  rise  to  various  and  changeable 
distributions  of  light  on  the  eclipsed  disc ; '  while,  if  entirely  clear,  the 
eclipse  will  be  remarkable  for  the  conspicuousness  of  the  moon  during 
the  whole  or  a  part  of  its  immersion  in  the  umbra.' 

'  As  in  the  eclipses  of  June  5,  1620,  April  25,  1642.    Lalande,  Ast.  1769. 

*  As  in  the  eclipse  of  Oct.  13,  1837,  observed  by  the  author. 

'  As  in  that  of  March  19,  1848,  when  the  moon  is  described  as  giving  "  good  light" 
during  more  than  an  hour  after  its  total  immersion,  and  some  persons  even  doubted 
its  being  eclipsed.    'Notices  of  R.  Ast.  Soc.  viii.  p.  132.) 


LUNAR   SCLIPTIC   LIMITS. 


227 


(425.)  Owing  to  the  great  size  of  the  earth,  the  cone  of  its  umbra 
always  projects  far  beyond  the  moon;  so  that,  if,  at  the  time  of  a  lunar 
eclipse,  the  moon's  path  be  properly  directed,  it  is  sure  to  pass  through 
the  umbra.  This  is  not,  however,  the  case  in  solar  eclipses.  It  so 
happens,  from  the  adjustment  of  the  size  and  distance  of  the  moon,  that 
the  extremity  of  her  umbra  always  falls  near  the  earth,  but  sometimes 
attains  and  sometimes  fulls  short  of  its  surface.  In  the  former  case 
(represented  in  the  lower  figure  art.  420)  a  black  spot,  surrounded  by  a 
fainter  shadow,  is  formed,  beyond  which  there  is  no  eclipse  on  any  part 
of  the  earth,  but  within  which  there  may  be  either  a  total  or  partial  one, 
ag  the  spectator  is  within  the  umbra  or  penumbra.  When  the  apex  of 
the  umbra  falls  on  the  surface,  the  moon  at  that  point  will  appear,  for  an 
instant,  to  Just  cover  the  sun ;  but,  when  it  falls  short,  there  will  be  no 
total  eclipse  on  any  part  of  the  earth ;  but  a  spectator,  situated  in  or  near 
the  prolongation  of  the  axis  of  the  cone,  will  see  the  whole  of  the  mooa 
on  the  sun,  although  not  large  enough  to  cover  it,  i.  e.  he  will  witness  an 
annular  eclipse. 

(426.)  Owing  to  a  remarkable  enough  adjustment  of  the  periods  in 
which  the  moon's  si/nodical  revolution,  and  that  of  her  nodes,  are  per- 
formed, eclipses  return  after  a  certain  period,  very  nearly  in  the  same 
order  and  of  the  same  magnitude.  For  223  of  the  moon's  mean  synodi- 
cal  revolutions,  or  lunations,  as  they  are  called,  will  be  found  to  occupy 
6585-32  days,  and  nineteen  complete  synodical  revolutions  of  the  node  to 
occupy  6585-78.  The  difference  in  the  mean  position  of  the  node,  then, 
at  the  beginning  and  end  of  223  lunations,  is  nearly  insensible ;  so  that 
a  recurrence  of  all  eclipses  within  that  interval  must  take  place.  Accord- 
ingly, this  period  of  223  lunations,  or  eighteen  years  and  ten  days,  is  a 
very  important  one  in  the  calculation  of  eclipses.  It  is  supposed  to  have 
been  known  to  the  Chaldeans,  the  earliest  astronomers,  the  regular  return 
of  eclipses  having  been  known  as  a  physical  fact  for  ages  before  their 
exact  theory  was  understood.  In  this  period  there  occur  ordinarily  70 
eclipses,  29  of  the  moon  and  41  of  the  sun,  visible  in  some  part  of  the 
earth.  Seven  eclipses  of  either  sun  or  moon  at  most,  and  two  at  least 
(both  of  the  sun,)  may  occur  in  a  year. 

(427.)  The  commencement,  duration,  and  magnitude  of  a  lunar  eclipse 
are  much  more  easily  calculated  than  those  of  a  solar,  being  independent 
of  the  position  of  the  spectator  on  the  earth's  surface,  and  the  same  as  if 
viewed  from  its  centre.  The  common  centre  of  the  umbra  and  penumbra 
lies  always  in  the  ecliptic,  at  a  point  opposite  to  the  sun,  and  the  path 
described  by  the  moon  in  passing  through  it  is  its  true  orbit  as  it  etands 
at  the  moment  of  the  full  moon.     In  this  orbit,  its  position,  at  every 


■W\ 


•i    I 


, 


M 


228 


OUTLINES  OF  ASTRONOMY. 


iustant,  is  known  from  the  lunar  tables  and  ephemeris ;  and  all  we  have, 
therefore,  to  ascertain,  is,  the  moment  when  the  distance  between  the 
moon's  centre  and  the  centre  of  the  shadow  is  exactly  equal  to  the  sum 
of  the  semidiameters  of  the  moon  and  penumhraf  or  of  the  moon  and 
umbra,  to  know  when  it  enters  upon  and  leaves  them  respectively.  No 
Itmar  eclipse  can  take  place,  if,  at  the  moment  of  the  full  moon,  the  sun 
be  at  a  greater  angular  distance  from  the  node  of  the  moon's  orbit  than 
11°  21',  meaning  by  an  eclipse  the  inmersion  of  any  part  of  the  moon  in 
the  umbra,  as  its  contact  with  the  penumbra  cannot  bo  observed  (see  note 
to  art.  421). 

(428.)  The  dimensions  of  the  shadow,  at  the  place  where  it  crosses  the 
moon's  path,  require  us  to  know  the  distances  of  the  sun  and  moon  at  the 
time.  These  are  variable ;  but  are  calculated  and  set  down,  as  well  as 
their  semidiameters,  for  every  day,  in  the  ephemeris,  so  that  none  of  the 
data  are  wanting.  The  sun's  distance  is  easily  calculate^  irom  its  elliptic 
orbit;  but  the  moon's  is  a  mailer  of  more  difficulty,  hj  reason  of  the  prc- 
gresslve  motion  of  the  axis  of  the  lunar  orbit.  (Art.  409.) 

(429.)  The  physical  constitution  of  the  moon  is  better  known  to  us 
than  that  of  any  other  heavenly  body.  By  the  aid  of  telescopes,  we 
discern  inequalities  in  its  surface  which  can  •  be  no  other  than  mountains 
and  valleys, — for  this  plain  reason,  we  see  the  shadows  cast  by  the  former 
in  the  exact  proportion  as  to  length  which  they  ought  to  have,  when  we 
take  into  account  the  inclination  of  the  sun's  rays  to  that  part  of  the 
moon's  surface  on  which  they  stand.  The  convex  outline  of  the  limb 
turned  towards  the  sun  is  always  circular,  and  very  nearly  smooth ;  but 
the  opposite  border  of  the  enlightened  part,  which  (were  the  moon  a  per- 
fect sphere)  ought  to  be  an  exact  and  sharply  defined  ellipse,  is  always 
observed  to  be  extremely  ragged,  and  indented  with  deep  recesses  and  pro- 
minent points.  The  mountains  near  this  edge  cast  long  black  shadows, 
as  they  should  evidently  do,  when  we  consider  that  the  sun  is  in  the  act 
of  rising  or  setting  to  the  parts  of  the  moon  so  circumstanced.  But  as 
the  enlightened  edge  advances  beyond  them,  i.  e.  as  the  sun  to  them  gains 
altitude,  their  shadows  shorten ;  and  at  the  full  moon,  when  all  the  light 
falls  in  our  line  of  sight,  no  shadows  are  seen  on  any  part  of  her  surface. 
From  micrometrical  measures  of  the  lengths  of  the  shadows  of  the  more 
conspicuous  mountains,  taken  under  the  most  favourable  circumstances, 
the  heights  of  many  of  them  have  been  calculated.  Messrs.  Beer  and 
Maedler  in  the'r  elaborate  work,  entitled  "  Der  Mond,"  have  given  a  list 
cf  heights  resulting  from  such  measurements,  i'jr  no  less  than  1095  lunar 
mountains,  among  which  occur  all  degrees  of  elevation  up  tc  3569  toiscs, 
(22828  British  feet),  or  about  1400  feet  higher  than  Chimborazo  in  the 


d: 


PHYSICAL  CONSTITUTION   OF  THE   MOON. 


229 


Andes.  The  existence  of  such  mountains  is  further  corroborated  by  their 
appearance,  as  small  points  or  islands  of  light  beyond  the  extreme  edge 
of  the  enlightened  part,  which  are  their  tops  catching  the  sun-beams 
before  the  intermediate  plain,  and  which,  as  the  light  advances,  at  length 
connect  themselves  with  it,  and  appear  as  prominences  from  the  general 
edge.  .-y,  ,;■;«, ,^  ,„;.  ,,^'..  ■ .  ,.,,, ,  , 

(430.)  The  generality  of  the  lunar  mountains  present  a  striking  uni- 
formity and  singularity  of  aspect.  They  are  wonderfully  numerous, 
especially  towards  the  Southern  portion  of  the  disc,  occupying  by  far  the 
larger  portion  of  the  surface,  and  almost  universally  of  an  exactly  circu- 
lar or  cup-shaped  form,  foreshortened,  however,  into  ellipses  towards  the 
limb;  but  the  larger  have  for  the  most  part  flat  bottoms  within,  from 
which  rises  centrally  a  small,  steep,  conical  hill.  They  offer,  in  short,  in 
its  highest  perfection,  the  true  volcanic  character,  as  it  may  be  seen  in  the 
crater  of  Vesuvius,  and  in  a  map  of  the  volcanic  districts  of  the  Campi 
Phlegraei'  or  the  Puy  de  D6me,  but  with  this  remarkable  peculiarity, 
viz. :  that  the  bottoms  of  many  of  the  craters  are  very  deeply  depressed 
below  the  general  surface  of  the  moon,  the  internal  depth  being  of^^^en 
twice  or  three  times  the  external  height.  In  some  of  the  principal  ones, 
decisive  marks  of  volcanic  stratification,  arising  from  successive  deposits 
of  ejected  matter,  and  evident  indications  of  lava  currents  streaming 
outwards  in  all  directions,  may  be  clearly  traced  with  powerful  telescopes. 
(See  PI.  V.  fig.  2.*)  In  Lord  Robse's  magnificent  reflector,  the  flat 
bottom  of  the  crater  called  Albategnius  is  seen  to  be  strewed  with  blocks 
not  visible  in  inferior  telescopes,  while  the  exterior  of  another  (Aristillus) 
is  all  hatched  over  with  deep  gullies  radiating  towards  its  centre.  What  is, 
moreover,  extremely  singular  in  the  geology  of  the  moon  is,  that,  ■  Ithough 
nothing  having  the  character  of  seas  can  be  traced,  (for  the  dusiiy  spots, 
which  are  commonly  called  seas,  when  closely  examined,  present  appear- 
ances incompatible  with  the  supposition  of  deep  water,)  yet  there  aro 
large  regions  perfectly  level,  and  apparently  of  a  decided  alluvial  cha- 
racter. 

(431.)  The  moon  has  no  clouds,  nor  any  other  decisive  indications  of 
an  atmosphere.  Were  there  any,  it  could  not  fail  to  be  perceived  in  the 
occultations  of  stir.'  and  the  phaenomena  of  solar  eclipses,  as  well  as  in 
a  great  variety  of  other  phajnomena.  The  moon's  diameter,  for  exaiLple, 
as  measured  micrometrically,  and  as  estimated  by  the  interval  between 
the  disappearance  and  reappearance  of  a  star  in  an  occultation,  ought  to 
differ  by  twice  the  horizontal  refraction  at  the  moon's  surface.     No  appro- 


'  See  Breislak's  map  of  the  environs  of  Naples,  and  Desnaarest's  of  Auvergne. 
'  From  a  drawing  taken  with  a  reflector  of  twenty  feet  focal  length  (A.) 


280 


OUTLINES  OP  ASTRONOMY. 


ciable  differeoco  being  perceived,  we  are  entitled  to  conclude  the  non- 
existence of  any  atmosphere  dense  enough  to  cause  a  refraction  of  1"  i.  e, 
having  one  1980th  part  of  the  density  of  the  earth's  atmosphere.  In  a 
solar  eclipse,  the  existence  of  any  sensible  refracting  atmosphere  in  the 
moon,  would  enable  us  to  trace  the  limb  of  the  latter  beyond  the  cusps, 
externally  to  the  sun's  disc,  by  a  narrow,  hut  hrilUant  line  of  light, 
extending  to  some  distance  along  its  edge.  No  such  phaenomenon  is 
seen.  Very  faint  stars  ought  to  be  extinguished  before  occultation,  were 
any  appreciable  amount  of  vapour  suspended  near  the  surface  of  the  moon. 
But  such  is  not  the  case ;  when  occulted  at  the  bright  edge,  indeed,  the 
light  of  the  moon  extinguishes  small  stars,  and  even  at  the  dark  limb, 
the  glare  in  the  sky  caused  by  the  near  presence  of  the  Tuoon,  rendem 
the  occultation  of  very  minute  s^ars  unobst.  vable.  But  during  the  con- 
tinuance of  a  total  lunar  eclipse,  stars  of  the  tenth  and  eleventh  magni- 
tude are  seen  to  come  up  to  the  limb,  and  undergo  sudden  extinction  as 
well  as  those  of  greater  brightness.'  Hence,  the  climate  of  the  moon 
must  be  very  extraordinary;  the  alternation  being  that  of  unmitigated 
and  burning  sunshine  fiercer  than  an  equatorial  noon,  continued  for  a 
whole  fortnight,  and  the  keenest  severity  of  frost,  far  exceeding  that  of 
our  polar  winters,  for  an  equal  time.  Such  a  disposition  of  things  must 
produce  a  constant  transfer  of  whatever  moisture  may  exist  on  'ts  surface, 
from  the  point  beneath  the  sun  to  that  opposite,  by  distillation  in  vacuo 
after  the  manner  of  the  little  instrument  called  a  cryophorus.  The  con- 
sequence must  be  absolute  aridity  below  the  vertical  sun,  constant  accre- 
tion of  hoar  frost  in  the  opposite  region,  and,  perhaps,  a  narrow  zone  of 
running  water  at  the  borders  of  the  enlightened  ixemisphere.'  It  ia 
possible,  then,  that  evaporation  on  the  one  hand,  and  condensation  on  the 
other,  may  to  a  certain  extent  preserve  an  equilibrium  of  temperature, 
and  mitigate  the  extreme  severity  of  both  climates;  but  this  process, 
which  would  imply  the  continual  generation  and  destruction  of  an  atmo- 
sphere of  aqueous  vapour,  must,  in  conformity  with  what  has  been  said 
above  of  a  lunar  atmosphere,  be  confined  within  very  narrow  limits. 

(432.)  Though  the  surface  of  the  full  moon  exposed  to  us,  must  neces- 
sarily be  very  much  heated, — possibly  to  a  degree  much  exceeding  that  of 
boiling  water, —  yet  we  /eel  no  heat  from  it,  and  even  in  the  focus  of  large 
reflectors,  it  fails  to  afiect  the  thermometer.  No  doubt,  therefore,  its  heat 
(conformably  to  what  has  been  observed  of  that  of  bodies  heated  below 
the  point  of  lujiinosity)  is  much  more  readily  absorbed  in  traversing 
transparent  media  than  direct  solar  heat,  and  is  extinguished  in  the  upper 

■  '  .'<  'As  observed  by  myself  in  the  eclipse  of  Oct.  13,  1837. 

*  So  in  ed.  of  1833. 


CLIMATE  AND   HEAT   OF  THE   MOON. 


231 


regions  of  our  atmosphere,  never  rcachint;  the  surface  of  the  earth  at  all. 
Some  probability  is  given  to  this  by  tue  tendency  to  disappearance 
of  clouds  under  the  full  moon,  a  meteorological  fact,  (for  as  f  uch  wo 
think  it  fully  entitled  to  rank')  for  which  it  is  necessary  to  seek  a  cause, 
and  for  which  no  other  rational  explanation  seems  to  offer.  As  for  any 
other  influence  of  the  moon  on  the  weather,  we  have  nc  decisive  evidence 
in  its  favour.  'i'»;.ji-f  ;ii  <^ir»v<^-'iJ  -^  H  r-  'i-  :  ^r,,  .1  '■■■>   ,';..;>  ;-;;.    w' .; 

(433.)  A  circle  of  one  second  in  diameter,  as  seen  from  the  earth,  on 
the  suiface  of  the  moon,  contains  about  a  square  mile.  Telescopes,  there- 
fore, must  yet  be  greatly  improved,  before  we  could  expect  to  see  signs  of 
inhabitants,  as  manifested  by  edifices  or  by  changes  on  the  surface  of  the 
floil.  It  should,  however,  be  observed,  that,  owing  to  the  small  density 
of  the  materials  of  the  moon,  and  the  comparatively  feeble  gravitation  of 
bodies  on  her  surface,  muscular  force  would  there  go  six  times  as  far  in 
overcoming  the  weight  of  materials  as  on  the  earth.  Owing  to  the  want 
of  air,  however,  it  seems  impossible  that  any  form  of  life,  analogous  to 
those  on  earth,  can  subsist  there.  No  appearance  indicating  vegetation, 
or  the  slightep*  variation  of  surface,  which  can,  in  our  opinion,  fairly  be 
ascribed  to  change  of  season,  can  any  where  be  discerned. 

(434.)  The  lunar  summer  and  winter  arise,  in  fact,  from  the  rotation 
of  the  moon  on  its  own  axis,  tl  3  period  of  which  rotation  is  exactly  equal 
to  its  sidereal  revolution  about  the  earth,  and  is  performed  in  a  plane  1° 
30'  11"  inclined  to  the  ecliptic,  whose  ascending  node  is  always  precisely 
coincident  with  the  descending  node  of  the  lunar  orbit.  So  that  the  axis 
of  rotation  describes  a  conical  surface  about  the  pole  of  the  ecliptic  in 
one  revolution  of  the  node.  The  remarkable  coincidence  of  the  two  rota- 
tions, that  about  the  axis  and  that  about  the  earth,  which  at  first  sight 
would  seem  perfectly  distinct,  has  been  asserted  (but  we  think  somewhat 
too  hastily')  to  be  a  consequence  of  the  general  laws  to  be  explained  here- 
after. Be  that  as  it  may,  it  is  the  cause  why  we  always  see  the  same  face 
of  the  moon,  and  have  no  knowledge  of  the  other  side. 

(435.)  The  moon's  rotation  on  her  axis  is  uniform;  but  since  her 
motion  ia  her  orbit  (like  that  of  the  sun)  is  not  so,  w'>  are  enabled  to 
look  a  few  degrees  round  the  equatorial  parts  of  her  visible  border,  on  the 
eastern  or  western  side,  according  to  circumstances;  or,  in  other  words, 
the  line  joining  the  centres  of  the  earth  and  moon  fluctuates  a  little  in  its 
position,  from  its  mean  or  averag    intersection  with  her  surface,  to  the 

'  From  my  own  observation,  made  quite  independently  of  any  knowledge  of  such 
a  tendency  having  been  observed  by  others.  Humboldt,  however,  in  his  personal  nar- 
rative, speaks  of  it  as  well  known  to  the  pilots  and  seamen  of  Spanish  America :  see 
note  at  the  end  of  the  chapter  (h.) 

•See  Edinburgh  Review,  No.  175,  p.  192. 


i  ;:  ) 


! 


OUTLINES   OF  ASTRONOMY. 


m 

m 


cast  or  westward.  And,  moreover,  since  the  axis  about  which  she  revolves 
Ls  neither  exactly  perpendicular  to  her  orbit,  nor  holds  an  invariable  direc- 
tion in  space,  her  poles  come  alternately  into  view  for  a  small  space  at  the 
edges  of  her  disc.  These  phenomena  are  known  by  the  name  of  Ubratiom 
In  consequence  of  these  two  distinct  kinds  of  libration,  the  same  identi- 
cal point  of  the  moon's  surface  is  not  always  the  centre  of  her  disc,  and 
we  therefore  get  sight  of  a  zone  of  a  few  degrees  in  breadth  on  all  sides 
of  the  border,  beyond  an  exact  hemisphere. 

(436.)  If  there  be  inhabitants  in  the  moon,  the  earth  must  present  to 
them  the  extraordinary  appearance  of  a  moon  of  nearly  2°  degrees  in 
diameter,  exhibiting  phages  complementary  to  those  which  we  see  the 
moon  to  do,  but  immoveable  Jixed  in  their  sky,  (or,  at  least,  changing  its 
apparent  place  only  by  the  small  amount  of  the  libration,)  while  the  stars 
must  seem  to  pass  slowly  beside  and  behind  it.  It  will  appear  clouded 
with  variable  spots,  and  belted  with  equatorial  and  tropical  zon<;S  corres- 
ponding to  our  trade-winds ;  and  it  may  be  doubted  whether,  in  their  per- 
petual change,  the  outlines  of  our  continents  and  seas  can  ever  be  clesirl^ 
discerned.  During  a  solar  eclipse,  the  earth's  atmosphere  will  become 
visible  as  a  narrow,  but  bright  luminous  ring  of  a  ruddy  colour,  where  it 
rests  on  the  earth,  gradually  passing  into  faint  blue,  encircling  the  whole 
or  port  of  the  dark  disc  of  the  earth,  the  remainder  being  dark  and  rugged 
with  clouds. 

(437.)  The  best  charts  of  the  lunar  surface  are  those  of  Cassini,  of 
Russel  (engraved  from  drawings,  made  by  the  aid  of  a  seven  feet  reflect- 
ing telescope,)  the  seleno-topographical  charts  of  Lohrmann,  and  the  very 
elaborate  projection  of  Beer  and  Maedler  accompanying  their  work 
already  cited.'  Madame  Witte,  a  Hanoverian  lady,  has  recently  suc- 
ceeded in  producing  from  her  own  observations,  aided  by  Maedlar's 
charts,  more  than  one  complete  model  of  the  whole  visible  lunar  hemi- 
sphere, of  the  most  perfect  kind,  the  result  of  incredible  diligence  and 
assiduity.  Single  craters  have  also  been  modelled  ca  a  large  scale,  both 
by  her  and  Mr.  Nasmyth.  [Still  more  recently  (1851)  photography  has 
been  successfully  applied  to  the  exact  delineation  of  the  lunar  surface  by 
Mr.  Whipple,  using  for  the  purpose  the  great  Fraanhorier  equatorial  of 
the  Observatory  at  Cambridge,  U.  S.]  ";    ..    "it    r 

*  The  representations  of  Heveliiis  in  his  Selenographia,  though  not  without  great 
merit  at  the  time,  and  fine  specimens  of  his  own  engraving,  are  now  become  antiquated. 

Additional  Note  on  Art.  ^'aZ 

M.  Arago  has  shown,  from  a  comparison  of  rain,  registered  as  having  fallen  during 
a  long  period,  that  a  slight  preponderance  in  respect  of  quantity  falis  near  the  new 
Moon  over  that  which  falls  near  the  full.  This  would  be  a  natural  and  necessary  con- 
sequence of  a  preponderance  of  a  cloudless  s'.y  about  the  full,  and  forms,  therefore, 
part  and  parcel  of  the  same  meteorological  fact. 


OF  TERRK8TIAL  GRAVITY. 


288 


I      ,1  'V  t, 


CHAPTER  Vni. 


OF  TERRESTRIAL  GRAVITY. 


TION.  —  PATHS  OP  PROJECTILES  J    APPARENT  —  REAL 


OP  THE  LAW  OP  UNIVERSAL  GRAVITA- 

TIIE   MOON 

RETAINED  IN  HER  ORBIT  BY  GRAVITY. — ITS  LAW  OP  DIMINUTION. — 
LAWS  OP  ELLIPTIC  MOTION.  —  ORBIT  OP  THE  EARTH  ROUND  THE  SUN 
IN  ACCORDANCE  WITH  THESE  LAWS.  —  MASSES  OP  THE  EARTH  AND 
SUN  COMPARED. — DENSITY  OP  THE  SUN. — FORCE  OP  GRAVITY  AT  ITS 
SURFACE. — DISTURBING  EFFECT  OP  THE  SUN  ON  THE  MOON's  MOTION. 

(438.)  The  reader  has  now  been  made  acquainted  with  the  chief  phe- 
nomena of  the  motions  of  the  earth  in  its  orbit  round  the  sun,  and  of  the 
moon  about  the  earth. — We  come  next  to  speak  of  the  physical  cause 
which  maintains  and  perpetuates  these  motions,  a.id  causes  the  massive 
bodies  so  revolving  to  deviate  continually  from  the  directions  they  would 
naturally  seek  to  follow,  in  pursuance  of  the  first  law  of  motion,'  and 
bend  their  courses  into  curves  concave  to  their  centres. 

(439.)  Whatever  attempts  may  have  been  made  by  metaphysical 
writers  to  reason  away  the  connection  of  cause  and  eflfect,  and  fritter  it 
down  into  the  unsatisfactory  relation  of  habitual  sequence,*  it  is  certain 
that  the  conception  of  some  more  real  and  intimate  connection  is  quite  as 
strongly  impressed  upon  the  human  mind  as  that  of  the  esisitence  of  an 
external  world, — the  vindication  of  whose  reality  has  (strange  to  say) 
been  regarded  as  an  achievement  of  no  common  merit  in  the  annals  of 
this  branch  of  philosophy.  It  is  our  own  immediate  consciousness  of 
effort,  when  we  exert  force  to  put  matter  in  motion,  or  to  oppose  and  neu- 
tralize force,  which  gives  us  this  internail  conviction  of  power  and  causa- 

'  Princip.  Lex.  i. 

*  See  Brown  '•  On  Cause  and  Effect,"  —  a  v/ork  of  great  acutcness  and  subtlety  of 
reasoning  on  some  points,  but  in  which  the  whole  train  of  argument  is  vitiated  by  one 
enormous  oversight ;  the  omission,  namely,  of  a  diatinet  and  immediate  personal  con- 
iciousneis  of  eauiation  in  his  enumeration  of  that  sequence  of  events,  by  which  the 
volition  of  the  mind  is  made  to  terminate  in  the  motion  of  material  objects.  I  mean 
the  consciousness  of  effort,  accompanied  with  intention  thereby  to  accomplish  an  end, 
as  a  thing  entirely  distinct  from  mere  desire  or  volition  on  the  one  hand,  and  from  mere 
spasmodic  contraction  of  muscles  on  the  other.  Brown,  3d  edit.  Edin.  1818,  p.  47. 
(Note  to  edition  of  1833.) 


•  'i  If  I 


234 


OUTLINES   OP   ASTRONOMY. 


tion  80  far  ns  it  refers  to  the  material  world,  and  compels  us  to  believe 
that  whenever  we  see  material  objeels  put  in  motion  from  a  state  of  rest, 
or  deflected  from  their  rectilinear  paths  and  changed  in  their  velocities  if 
already  in  motion,  it  is  in  consequence  of  such  an  effort  somehow 
exerted,  though  not  accompanied  with  our  consciousness.  That  such  an 
effort  should  be  exerted  with  success  through  an  interposed  space,  is  no 
more  difficult  to  conceive,  than  that  our  hand  should  communicate  motion 
to  a  stone,  with  which  it  is  demonatrahli/  not  in  contact. 

(440.)  All  bodies  with  which  we  are  acquainted,  when  raised  into  the 
air  and  quietly  abandoned,  descend  to  the  earth's  surface  in  lines  perpen- 
dicular to  it.  They  are  therefore  urged  thereto  by  a  force  or  effort,  which 
it  is  but  reasonable  to  regard  as  the  direct  or  indirect  result  of  a  conscious- 
ness and  a  will  existing  somewhere,  though  beyond  our  power  to  trace, 
which  force  we  term  gravity,  and  whose  tendency  or  direction,  as  uni- 
versal experience  teaches,  is  towards  the  earth's  centre;  or  rather,  to 
speak  strictly,  with  reference  to  its  spheroidal  figure,  perpendicular  to  the 
surface  of  still  water.  But  if  we  cast  a  body  obliquely  into  the  air,  thij 
tendency,  though  not  extinguished  or  diminished,  is  materially  modified 
in  its  ultimate  effect.  The  upward  impetus  we  give  the  stone  is,  it  is 
true,  after  a  time  destroyed,  and  a  downward  one  communicated  to  it, 
which  ultimately  brings  it  to  the  surface,  where  it  is  opposed  in  its  fur- 
ther jrogress,  and  brought  to  rest.  But  all  the  while  it  has  been  conti- 
nually deflected  or  bent  aside  from  its  rectilinear  progress,  and  made  to 
describe  a  curved  lino  concave  to  the  earth's  centre  j  and  having  a  highest 
point,  vertex,  or  apogee,  just  as  the  moon  has  in  its  orbit,  where  the  direc- 
tion of  its  motion  is  perpendicular  to  the  radius. 

(441.)  When  the  stone  which  we  fling  obliquely  upwards  meets  and  is 
stopped  in  its  descent  by  the  earth's  surface,  its  motion  is  not  towards  ike 
centre,  but  inclined  to  the  earth's  radius  at  the  same  angle  as  when  it 
quitted  our  hand.  As  we  are  sure  that,  if  not  stopped  by  the  resistance 
of  the  earth,  it  would  continue  to  descend,  and  that  ohliquely,  what  pre- 
sumption, we  may  ask,  is  there  that  it  would  ever  reach  the  centre  towards 
which  its  motion,  in  no  part  of  its  visible  course,  was  ever  directed  t 
What  reason  have  we  to  believe  that  it  might  not  rather  circulate  round 
it,  as  the  moon  does  round  the  earth,  returning  again  to  the  point  i*  set 
out  from,  after  completing  an  elliptic  orbit  of  which  the  earth's  cc.\tre 
occupies  the  lower  focus  ?  And  if  so,  is  it  not  reasonable  to  imagine  that 
the  same  force  of  gravity  may  (since  we  know  that  it  is  exerted  at  all 
accessible  heights  above  the  surface,  and  even  in  the  highest  regions  of 
the  atmosphere)  extend  as  far  as  60  radii  of  the  earth,  or  to  the  moon? 
and  may  not  this  be  the  power, — for  som^  power  there  mvst  be, — which 


GRAVITATION  OF  THB  MOON  TO  THE  EARTH. 


2S5 


deflects  her  at  every  instant  from  the  tangent  of  her  orbit,  and  keeps  her 
in  the  elliptic  path  which  experience  teaches  us  she  actually  pursues  ?    '    ' 

(442.)  If  a  stone  be  whirled  round  at  the  end  of  a  string  it  will  stretch 
the  string  by  a  centrifugal  force,  which,  if  the  speed  of  rotation  be  suffi< 
oiently  increased,  will  at  length  break  the  string,  and  let  the  stone  escape. 
However  strong  the  string,  it  may,  by  a  sufficient  rotary  velocity  of  the 
stone,  be  brought  to  the  utmost  tension  it  will  bear  without  breaking;  and 
if  we  know  what  weight  it  is  capable  of  carrying,  the  velocity  necessary 
for  this  purpose  is  easily  calculated.  Suppose,  now,  a  string  to  connect 
the  earth's  centre  \t'ith  a  weight  at  its  su/faco,  whose  strength  should  be 
just  sufficient  to  sustain  that  weight  suspended  from  it.  Let  us,  however, 
for  a  moment  imagine  gravity  to  have  no  ejcistence,  and  that  the  weight 
is  made  to  revolve  with  the  limiting  vdocity  which  that  string  can  barely 
counteract :  then  will  its  tension  be  just  equal  to  the  weight  of  the  rc< 
volving  body;  and  any  power  which  should  continually  urge  the  body 
towards  the  centre  with  a  force  equal  to  its  weight  would  perform  the 
office,  and  might  supply  the  place  of  the  string,  if  divided.  Divide  it 
then,  and  in  its  place  let  gravity  ac^,  and  the  body  will  circulate  as  before ; 
its  tendency  to  the  centre,  or  its  weight,  being  just  balanced  by  its  centri- 
fugal force  Knowing  the  radius  of  the  earth,  we  can  calculate  by  tho 
princif)les  of  mechanics  the  periodical  time  in  which  a  body  so  balanced 
must  circulate  to  keep  it  up;  and  this  appears  to  be  1"  23">  22*. 

(443.)  If  we  make  the  same  calculation  for  a  body  at  the  distance  of 
the  moon,  supposing  its  weight  or  gravity  the  same  as  at  the  earth's 
iurface,  we  shall  find  tho  period  required  to  be  10"  45"  30».  The  actual 
period  of  the  moon's  revolution,  however,  is  27'  7"  43" ;  and  hence  it  is 
clear  that  the  moon's  velocity  is  not  nearly  sufficient  to  sustain  it  against 
mch  a  power,  supposing  it  to  revolve  in  a  circle,  or  neglecting  (for  the 
present)  the  slight  ellipticity  of  its  orbit.  In  order  that  a  body  at  the 
distance  of  the  moon  (or  the  moon  itself)  should  be  capable  of  keeping 
its  distance  from  the  earth  by  the  outward  eflFort  of  its  centrifugal  force, 
while  yet  its  time  of  revolution  should  be  what  the  moon's  actually  is, 
it  will  appear '  that  gravity,  instead  of  being  as  intense  as  at  the  surface, 
would  require  to  be  very  nearly  3600  times  less  energetic ;  or,  in  other 
words,  that  its  intensity  is  so  enfeebled  by  the  remoteness  of  the  body  on 
ivhich  it  acts,  as  to  be  capable  of  producing  in  it,  in  the  same  time,  only 
s^'^^tb  part  of  the  motion  which  it  would  impart  to  the  same  mass  of 
matter  at  the  earth's  surface. 

(444.)  The  distance  of  the  moon  from  the  earth's  centre  is  a  very  little 
less  than  sixty  times  the  distance  from  the  centre  to  the  surface,  and 

'  Nowton,  Princip.  b.  i.,  Prop.  4.,  Cor.  2. 


280 


OI^TLINEB  OF  ASTRONOMY. 


8600  :  1  :  :  GO'  :  1';  so  tlmt  the  proportion  in  wbioh  we  must  admit  the 
earth's  gravity  to  be  enfeebled  at  the  moon's  distance,  if  it  bo  rcully  the 
force  which  retains  the  moon  in  her  orbit,  must  be  (at  least  in  this  par- 
ticular instance)  that  of  the  squares  of  the  distances  at  which  it  is  com- 
pared. Now,  in  such  a  diminution  of  energy  with  increase  of  dihtuuco, 
there  is  nothing  prima  facie  inadmissible.  Emanations  from  a  centre, 
such  as  light  and  heat,  do  really  diminish  in  intensity  by  increase  of  dis- 
tance, and  in  this  identical  proportion ;  and  though  we  cannot  certainly 
argue  much  from  this  analogy,  yet  we  do  see  that  the  power  of  niagnetio 
and  electric  attractions  and  repulsions  is  actually  enfeebled  by  distaucc, 
and  much  more  rapidly  than  in  the  simple  proportion  of  the  increased 
distances.  The  argument,  therefore,  stands  thus  :  —  On  the  one  hand, 
Gravity  is  a  real  power,  of  whose  ogepcy  wc  have  daily  experience.  We 
know  that  it  extends  to  the  greatest  accessible  heights,  and  far  beyond ; 
and  we  sec  no  reason  for  drawing  a  line  at  any  particular  height,  and 
there  asserting  that  it  must  cease  entirely ;  though  we  have  analogies  to 
lead  us  to  suppose  its  energy  may  diminish  as  we  ascend  to  great  heights 
from  the  surface,  such  as  that  of  the  moon.  On  the  other  hand  wo  are 
sure  the  moon  is  urged  towards  the  earth  by  some  power  which  retains 
her  in  her  orbit,  and  that  the  intensity  of  this  power  is  such  as  would 
correspond  to  a  gravity,  diminished  in  tho  proportion  —  otherwise  not 
improbable  —  of  tho  squares  of  the  distances.  If  gi'avity  be  not  that 
power,  there  must  exist  some  other;  and,  besides  this,  gravity  must 
cease  at  some  inferior  level,  or  the  nature  of  tho  moon  must  be  different 
from  that  of  ponderable  matter ;  —  for  if  not,  it  would  be  urged  by  both 
powers,  and  therefore  too  much  urged  and  forced  inwards  from  her  path. 

(445.)  It  is  on  such  an  argument  that  Newton  is  understood  to  have 
rested,  in  the  first  instance,  and  provisionally,  his  law  of  universal  gravi- 
tation, which  may  be  thus  abstractly  stated : — '<  Every  particle  of  matter 
in  the  universe  attracts  every  other  particle,  with  a  force  directly  propor- 
tioned to  the  mass  r''  the  attracting  particle,  and  inversely  to  the  square 
of  the  distance  between  them."  In  this  abstract  and  general  form, 
however,  the  propo«itiv«  is  not  •pplicablo  to  the  case  before  us.  The 
earth  and  moon  are  oot  mere  particles,  but  great  spherical  bodies,  and  to 
such  the  general  law  does  not  immediately  apply  j  and,  before  we  can 
make  it  applicable,  it  becomes  necessary  to  inquire  what  will  be  the  force 
with  which  a  congerien  of  particles,  constituting  a  solid  mass  of  any 
assigned  figure,  will  attract  c  .&«r  such  collection  of  material  atoms. 
This  problem  is  one  purely  dyaaaaical,  and,  in  this  its  general  forr.i,  is 
of  oxtrem<:  Hfficulty  Fortunately  however,  for  human  knowledge,  wh.  : 
the  attrac    ^  and  attracted  bodies  are  spheres,  it  adiaita  of  an  eu^y  ana 


GENERAL  LAW   OP  GRAVITATION. 


28T 


i   ',: 


direct  solution.  Newton  himself  has  shown  (Princip.  b.  i.  prop.  75) 
that,  in  that  case,  tho  attraction  is  precisely  the  same  as  if  the  whole 
laattcr  of  each  sphere  wore  collected  into  its  centre,  and  the  spheres  were 
Binglo  particles  there  placed ;  so  that,  in  this  case,  the  general  law  applies 
in  its  strict  wording.  The  effect  of  the  trifling  deviation  of  tho  earth 
from  a  spherical  form  is  of  too  minute  an  order  to  need  attention  at  pre- 
sent.    It  is,  however,  perceptible,  and  may  be  hereafter  noticed. 

(440.)  The  next  step  in  the  Newtonian  argument  is  one  which  divests 
the  law  of  gravitation  of  its  provisional  character,  as  derived  from  a  loose 
and  superficial  consideration  of  tho  lunar  orbit  as  a  circlo  described  with  an 
average  or  mean  velocity,  and  elevates  it  to  the  rank  of  a  general  and  pri- 
mordial relation  by  proving  its  applicability  to  the  state  of  existing  nature 
in  all  its  circumstances.  This  step  consists  in  demonstrating,  as  ho  has 
done '  (^Princip.  i.  17.  i.  75.),  that,  under  the  influence  of  such  an  attract- 
ive force  mutually  urging  two  spherical  gravitating  bodies  towards  each 
other,  they  will  each,  when  moving  in  each  other's  neighbourhood,  be 
deflected  into  an  orbit  concave  towards  tho  other,  and  describe,  one  about 
the  other  regarded  as  fixed,  or  both  round  their  common  centre  of  gravity, 
curves  whose  forms  are  limited  to  those  figures  known  in  geometry  by  tho 
general  name  of  conic  sections.  It  will  depend,  ho  shows,  in  any  assigned 
case,  upon  the  particular  circumstances  or  velocity,  distance,  and  direction, 
wAu7t  of  these  curves  shall  bo  described,  —  whether  an  ellipse,  a  circlo,  a 
parabola,  or  an  hyperbola  j  but  one  or  other  it  must  be ;  and  any  one  of 
any  degree  of  excentricitv  '\\  mni/  be,  according  to  the  circumstances  of 
the  case ;  and,  in  all  e»si.>3,  the  point  to  which  the  motion  is  referred, 
whether  it  be  the  ceurtow  of  one  of  the  spheres,  or  their  common  centre  of 
gravity,  will  of  necessity-  be  the  focus  of  the  come  section  described.  He 
Bhows,  furthermore  (^Princip.  i.  1.),  that,  in  every  case,  the  angular  velo- 
city with  which  the  line  joining  their  centres  moves,  must  be  inversely 
proportional  to  the  square  of  their  mutual  distance,  and  that  equal  areas 
of  the  curves  described  will  be  swept  over  by  their  line  of  junction  in 
equal  times. 

(447.)  All  this  is  in  conformity  with  what  we  have  stated  of  the  solar 

'  We  refer  for  these  fundamental  propositions,  as  a  point  of  duty,  to  the  immortal 
\vork  in  vrtiich  they  were  first  propounded.  It  is  impossible  for  us,  in  this  volume,  tu 
go  into  these  investigations:  even  did  our  limits  permit,  it  would  be  utterly  inconsist- 
ent with  our  plan  :  a  general  idea,  however,  of  their  conduct  will  be  given  in  the  next 
chapter.  We  trust  that  the  careful  and  attentive  study  of  the  Principia  in  its  original 
form  will  never  be  laid  aside,  whatever  be  the  improvements  of  the  modern  analysis 
as  u?pects  facility  of  calculation  and  expression.  From  no  other  quarter  can  a  thorough 
and  complete  comprehension  of  the  mechanism  of  our  system,  (so  far  as  the  immedi- 
ate scope  of  that  work,)  be  anything  like  so  well,  and  we  may  add,  so  easily  obtained. 


238 


OUTLINES   OF  ASTRONOMY. 


II 


I 

i 


and  lunar  movements.  Their  orbits  are  ellipses,  but  of  different  d<?grees 
of  excentricity ;  and  this  circumstance  already  indicates  the  general  appli. 
cability  of  the  principles  in  question. 

(448.)  But  here  we  have  already,  by  a  natural  and  ready  implication 
(such  is  always  the  progress  of  generalization),  taken  a  further  and  most 
important  step,  almost  unperceived.  We  have  extended  the  action  of 
gravity  to  the  case  of  the  earth  and  sun,  to  a  distance  immensely  greater 
than  that  of  the  moon,  and  to  a  body  apparently  quite  of  a  different 
nature  from  either.  Are  we  justified  in  this?  or,  at  all  events,  are  there 
no  modifications  introduced  by  the  change  of  data,  if  not  into  the  general 
expression,  at  least  into  the  particular  interpretation,  of  the  law  of  gravi- 
tation ?  Now,  the  moment  we  come  to  numbers,  an  obvious  incongruity 
strikes  us.  When  we  calculate,  as  above,  from  the  known  distance  of  the 
sun  (art.  357),  and  from  the  period  in  which  the  earth  circulates  about  it 
(art.  305),  what  must  be  the  centrifugal  force  of  the  latter  by  which  the 
sun's  attraction  is  balanced,  (and  which,  therefore,  becomes  an  exact  mea- 
sure of  the  sun's  attractive  energy  as  exerted  on  the  earth,)  we  find  it  to 
be  immensely  greater  than  would  suflBce  to  counteract  the  eartJi's  attrac- 
tion on  an  equal  body  at  that  distance — greater  in  the  high  proportion  of 
354936  to  1.  It  is  clear,  then,  that  if  the  earth  be  retained  in  its  orbit 
about  the  sun  by  solar  attraction,  conformable  in  its  rate  of  diminution 
with  the  general  law,  this  force  must  be  no  less  than  354936  times  more 
intense  than  what  the  earth  would  be  capable  of  exerting,  cseteris  paribus, 
at  an  equal  distance. 

(449.)  What,  then,  are  we  to  understand  from  this  result  ?  Simply 
this, —  that  the  sun  attracts  as  a  collection  of  354936  earths  occupying 
its  place  would  do,  or,  in  other  words,  that  the  sun  contains  354936  times 
the  mass  or  quantity  of  ponderable  matter  that  the  earth  consists  of.  Nor 
let  this  conclusion  startle  us.  We  have  only  to  recall  what  has  been 
already  shown  (in  art.  358)  of  the  gigantic  dimensions  of  this  magnifi- 
cent body,  to  perceive  that,  in  assigning  to  it  so  vast  a  mass,  we  are  not 
outstepping  a  reasonable  proportion.  In  fact,  when  we  come  to  compare 
its  mass  with  its  bulk,  we  find  its  density'  to  be  less  than  that  of  the  earth, 
being  no  more  than  0-2543.  So  that  it  must  consist,  in  reality,  of  far 
liyhter  materials,  Qspecially  when  we  consider  the  force  under  which  its 
central  parts  must  be  condensed.  This  consideration  renders  it  highly 
probable  that  an  intense  heat  prevails  in  its  interior  by  which  its  elasticity 


moon,  while  in 
their  common 


*  The  density  of  a  material  body  is  as  the  masi  directly,  and  the  volume  inversely; 
hence  density  of  O :  density  of  ® : :  j^ff  J?| :  1  :  02543  :  1. 


GRAVITATION   AND   MASS   OF  THE  SUN. 


239 


ia  reinforced,  and  rendered  capable  of  resisting  this  almost  inconceivable 
pressure  without  collapsing  into  smaller  dimensions. 

(450.)  This  will  be  more  distinctly  appreciated,  if  we  estimate,  as  we 
are  now  prepared  to  do,  the  intensity  of  gravity  at  the  sun's  surface. 

The  attraction  of  a  sphere  being  the  same  (art.  445)  as  if  its  whole 
mass  were  collected  in  its  centre,  will,  of  course,  be  proportional  to  the 
mass  directly,  and  the  square  of  the  distance  inversely ;  and,  in  thJs  ca^e, 
the  distance  is  the  radius  of  the  sphere.  Hence  we  conclude',  that  the 
intensities  of  solar  and  terrestrial  gravity  at  the  surfaces  of  the  two  globes 
are  in  the  proportions  of  27-9  to  1.  A  pound  of  terrestrial  matter  at  the 
sun's  surface,  then,  would  exert  a  pressure  equal  to  what  27"9  such  pounds 
would  do  at  the  earth's.  The  efficacy  of  muscular  power  to  overcome 
weight  is  therefore  proportionally  nearly  28  times  less  on  the  sun  than  on 
the  earth.  An  ordinary  man,  for  example,  would  not  only  be  unable  to 
sustain  his  own  weight  on  the  sun,  but  would  be  literally  crushed  to  atoms 
under  the  load.' 

(451.)  Henceforward,  then,  we  must  consent  to  dismiss  all  idea  of  the 
earth's  immobility^  and  transfer  that  attribute  to  the  sun,  whose  ponderous 
mass  is  calculated  to  exhaust  the  feeble  attractions  of  such  comparative 
atoms  as  the  earth  and  moon,  without  being  perceptibly  dragged  from  its 
place.  Their  centre  of  gravity  lies,  as  we  have  already  hinted,  almost 
close  to  the  centre  of  the  solar  globe,  at  an  interval  quite  imperceptible 
from  our  distance  j  and  whether  we  regard  the  earth's  orbit  as  being  per- 
formed about  the  one  or  the  other  centre  makes  no  appreciable  difference 
in  any  one  phenomenon  of  astronomy. 

(452.)  It  is  in  consequence  of  the  mutual  gravitation  of  all  the  several 
parts  of  matter,  which  the  Newtonian  law  supposes,  that  the  earth  and 
moon,  while  in  the  act  of  revolving,  monthly,  in  their  mutual  orbits  about 
their  common  centre  of  gravity,  yet  continue  to  circulate,  without  parting 
company,  in  a  greater  annual  orbit  round  the  sun.  We  may  conceive  this 
motion  by  connecting  two  unequal  balls  by  a  stick,  which,  at  their  centre 
of  gravity,  is  tied  by  a  long  string,  and  whirled  round.  Their  joint  si/S' 
tern  will  circulate  as  one  body  about  the  common  centre  to  which  the 
string  is  attached,  while  yet  they  may  go  on  circulating  round  each  other  in 
subordinate  gyrations,  as  if  the  stick  were  quite  free  from  any  such  tie, 
and  merely  hurled  through  the  air.  If  the  earth  alone,  and  not  the 
moon,  gravitated  to  the  sun,  it  would  be  dragged  away,  and  leave  the  moon 


3S4BS6        .  1 

(r?B0niT)2  •  (TTTiTIT) 


I  27'9  :  1 ;  the  respective  radii 


'  Solar  gravity :  terrestrial 
of  the  sun  and  earth  being  440000,  and  4000  miles. 

'A  mass  weighing  12  stone  or  1681bs.  on  the  earth,  would  produce  a  pressure  of 
4687  lbs.  on  the  sun. 


»  ■ 


A: 


240 


OUTLINES   OF  ASTRONOMY. 


bebind  —  and  vice  versd ;  but,  acting  on  both,  they  continue  together 
under  its  attraction,  just  as  the  loose  parts  of  the  earth's  surface  continue 
to  rest  upon  it.  ,  It  is,  then,  in  strictness,  not  the  earth  or  the  moon 
which  describes  an  ellipse  around  the  sun,  but  their  common  centre  of 
gravity.  The  eflFect  is  to  produce  a  small,  but  very  perceptible,  monthly 
equation  in  the  sun's  apparent  motion  as  seen  from  the  earth,  which  is 
always  taken  into  account  in  calculating  the  sun's  place.  The  moon's 
actual  path  in  its  compound  orbit  round  the  earth  and  sun  is  an  epicycloi- 
dal  curve  intersecting  the  orbit  of  the  earth  twice  in  every  lunar  month, 
and  alternately  within  and  without  it.  But  as  there  are  not  more  than 
twelve  such  months  in  the  year,  and  as  the  total  departure  of  the  moon 
from  it  either  way  does  not  exceed  one  400th  part  of  the  radius,  this 
amounts  only  to  a  slight  undulation  upon  the  earth's  ellipse,  so  slight, 
indeed,  that  if  drawn  in  true  proportion  on  a  large  sheet  of  paper,  no  eye 
unaided  by  measurement  with  compasses  would  detect  it.  The  real  orbit 
of  the  moon  is  everywhere  concave  towards  the  sun. 

(453.)  Here  moreover,  i.  e.  in  the  attraction  of  the  sun,  we  have  tiie 
key  to  all  those  differences  from  an  exact  elliptic  movement  of  the  moon 
in  her  monthly  orbit,  which  we  have  already  noticed  (arts.  407,  409),  viz. 
to  the  retrograde  revolution  of  her  nodes ;  to  the  direct  circulation  of  the 
axis  of  her  ellipse ;  and  to  all  the  other  deviations  from  the  laws  of  elliptic 
motion  at  which  we  have  further  hinted.  If  the  moon  simply  revolved 
about  the  earth  under  the  influence  of  its  gravity,  none  of  these  pheno- 
mena would  take  place.  Its  orbit  would  be  a  perfect  ellipse,  returning 
ihto  itself,  and  always  lying  in  one  and  the  same  plane.  That  it  is  not  so, 
is  a  proof  that  some  cause  disturbs  it,  and  interferes  with  the  earth's 
attraction ;  and  this  cause  is  no  other  than  the  sun's  attraction — or  rather, 
that  part  of  it  which  is  not  equally  exerted  on  the  earth. 

(454.)  Suppose  two  stones,  side  by  side,  or  otherwise  situated  with  re- 
spect to  each  other,  to  be  let  fall  together;  then,  as  gravity  accelerates 
them  equally,  they  will  retain  their  relative  positions,  and  fall  together  as 
if  they  formed  one  mass.  But  suppose  gravity  to  be  rather  more  intensely 
exerted  on  one  than  on  the  other ;  then  would  that  one  be  rather  more 
accelerated  in  its  faP,  and  would  gradually  leave  the  other ;  and  thus  a 
relative  motion  between  them  would  arise  from  the  difference  of  action, 
however  slight. 

(455.)  The  sun  is  about  400  times  more  remote  than  the  moon  j  and, 
in  consequence,  while  the  moon  describes  her  monthly  orbit  round  the 
earth,  her  distance  from  the  sun  is  alternately  ij^gth  part  greater  and  as 
much  less  than  the  earth's.  Small  as  this  is,  it  is  yet  sufficient  to  produce 
a  perceptible  excess  of  attractive  tendency  of  the  moon  towards  the  sun, 


SOLAR  DTSTUBANCB  OF  THE  MOON'S  SURFACE. 


241 


Fig.  62. 


NaM_ 


£ 


S 


above  that  of  the  earth  when  in  the  nearer  point  of  her  orbit,  M,  and  a 
corresponding  defect  on  the  opposite  part,  N ;  and,  in  the  intermediate 
positions,  not  only  will  a  difference  of  forces  subsist,  but  a  difference  of 
directions  also ;  since  however  small  the  lunar  orbit  M  N,  it  is  not  a  point, 
and,  therefore,  the  lines  drawn  from  the  sun  S  to  its  several  parts  cannot 
be  regarded  as  strictly  parallel.     If,  as  we  have  already  seen,  the  force  of 
the  sun  were  equally  exerted,  and  in  parallel  directions  on  both,  no  disturb- 
ance of  their  relative  siiuaiions  would  take  place ;  but  from  the  non-veri- 
fication of  these  conditions  arises  a  disturbing  force,  oblique  to  the  line 
joining  the  moon  and  earth,  which  in  some  situations  acts  to  accelerate,  in 
others  to  retard,  her  elliptic  orbitual  motion ;  in  some  to  draw  the  earth 
from  the  moon,  in  others  the  moon  firom  the  earth.     Again,  the  lunar 
orbit^  "--Mieih  very  nearly,  is  yet  not  quite  coincident  with  the  plane  of  the 
eclip         •  d  hence  the  action  of  the  sun,  which  is  very  nearly  parallel  to 
the  .  -   .  .outioned  plane,  tends  to  draw  her  somewhat  out  of  the  plane  of 
her  orbit,  and  does  actually  do  so — producing  the  revolution  of  her  nodes, 
and  other  phenomena  less  striking.     We  are  not  yet  prepared  co  go  into 
the  subject  of  these  perturbations,  as  they  are  called  j  but  they  are  intro- 
duced to  the  reader's  notice  as  early  as  possible,  for  the  purpose  of  re- 
assuring his  mind,  should  doubts  have  arisen  as  to  the  logical  correctness 
of  our  argument,  in  consequence  of  our  temporary  neglect  of  them  while 
working  our  way  upward  to  the  law  of  gravity  from  a  general  considera- 
tion of  the  moon's  orbit. 


V, 


16 


'! 


■y -'    %*^ 


242 


OUTLINES   OF  ASTRONOMT. 


I. 


■\  A 


CHAPTER  IX. 


OJ    THE    SOLAR    SYSTEM. 


|t 


APPARENT   MOTIONS    OP   THE    PLANETS.  —  THEIR   STATIONS    AND    RE- 
TR0GRADATI0N8.  —  THE   SUN   THEIR  NATURAL  CENTRE   OP  MOTION. 

—  INFERIOR  PLANETS.  —  THEIR  PHASES,  PERIODS,  ETC.  —  DIMEN- 
SIONS  AND   FORM   OP  THEIR  ORBITS.  —  TRANSITS  ACROSS   THE  SUN. 

—  SUPERIOR  PLANETS. — THEIR  DISTANCES,  PERIODS,  ETC.  —  KEP- 
LER's  LAWS  AND  THEIR  INTERPRETATION.  —  ELLIPTIC  ELEMENTS 
OP  A  planet's  orbit. — ITS  HELIOCENTRIC  AND  GEOCENTRIC 
PLACE.  —  EMPIRICAL  LAW  OP  PLANETARY  DISTANCES; — VIOLATED 
IN  THE  CASE  OP  NEPTUNE.  —  THE  ULTRA-ZODIACAL  PLANETS.— 
PHYSICAL  PECULIARITIES  OBSERVti-BLE  IN   EACH   OF  THE  PLANETS. 

(456.)  The  sun  and  moon  are  not  the  only  olestial  objects  which 
appear  to  have  a  motion  independent  of  that  by  which  the  great  con- 
stellation of  the  heavens  is  daily  carried  round  the  earth.  An^  ^ng  the 
stars  therj  are  several,  —  and  those  among  the  brightest  and  most  con- 
spicuous,—  which,  when  attentively  watched  from  night  to  night,  are 
found  to  change  their  relative  situations  among  the  rest ;  some  rapidly, 
others  much  more  slowly.  These  are  called  planets.  Four  of  them  — 
Venus,  Mars,  Jupiter,  and  Saturn  —  are  remarkably  large  and  brilliant; 
another.  Mercury,  is  also  visible  to  the  naked  eye  as  a  large  star,  but,  for 
a  reason  which  will  presently  appear,  is  seldom  conspicuous;  a  sixth, 
Uranus,  is  barely  discernible  without  a  telescope ;  and  nine  others — Nep- 
tune, Ceres,  Pallas,  Vesta,  Juno,  Aatraoa,  Hebe,  Iris,  Flora  —  are  never 
visible  to  the  naked  eye.  Besides  these  fifteen,  others  yet  undiscovered 
may  exist ;'  and  it  is  extremely  probable  that  such  is  the  case, — the  mul- 
titude of  telescopic  stars  being  so  great  that  only  a  small  fraction  of  tbeir 
number  has  been  suflSciently  noticed  to  ascertain  whether  they  retain  the 
same  places  or  not,  and  the  ten  last-mentioned  planets  having  all  been 
discovered  within  little  more  than  half  a  century  from  the  present  time. 

•  While  this  sheet  is  passing  through  the  press,  a  sixteenth,  not  yet  named,  has 
been  added  to  the  list,  by  the  observations  of  Mr.  Graham,  astronomical  assistant  to 
E.  Cooper,  Esq.  at  his  observatory  at  Markroe,  Sligo,  Ireland. 


APPARENT  MOTIONS  OF  THE  PLANETS. 


243 


'  f 


(457.)  The  apparent  motions  of  the  planets  are  much  more  irregular 
than  those  of  the  sun  or  moon.  Generally  speaking,  and  comparing  their 
places  at  distant  times,  they  all  advance,  though  mih  very  different  ave- 
rage  or  mean  velocities;  in  the  same  direction  as  those  luminaries,  i.  e.  in 
opposition  to  the  apparent  diurnal  motion,  or  from  west  to  east :  all  of 
them  make  the  entire  tour  of  the  heavens,  though  under  very  different 
circumstances;  and  all  of  them,  with  the  exception  of  the  eight  teles- 
copic planets, — Ceres,  Pallas,  June,  Vesta,  Astraea,  Hehe,  Iris,  and  Flora 
(which  may  therefore  be  termed  ultra-zodiacal^  —  are  confined  in  their 
visible  paths  within  very  narrow  limits  on  either  side  the  ecliptic,  and 
perform  their  movements  within  that  zone  of  the  heavens  we  have  called, 
above,  the  Zodiac  (art.  803.) 

(458.)  The  obvious  conclusion  from  this  is,  that  whatever  be,  other- 
wise, the  nature  and  law  of  their  motions,  they  are  performed  nearly  in 
the  plane  of  the  ecliptic  —  that  plane,  namely,  in  which  our  own  motion 
about  the  sun  is  performed.  Hence  it  follows,  that  we  see  their  evolu- 
tionp,  not  in  plan,  but  in  section;  their  real  angular  movements  and 
lineal  distances  being  all  foreshortened  and  confounded  undistinguishably, 
while  only  their  deviations  from  the  ecliptic  appear  of  their  natural  mag- 
nitude, undiminished  by  the  effect  of  perspective. 

(459.)  The  apparent  motions  of  the  sun  and  moon,  though  not  uni- 
form, do  not  deviate  very  greatly  from  uniformity ;  a  moderate  accelera- 
tion and  retardation,  accountable  for  by  the  ellipticity  of  their  orbits, 
being  all  that  is  remarked.  But  the  case  is  widely  different  with  the 
planets :  sometimes  they  advance  rapidly ;  then  relax  in  their  apparent 
speed  —  come  to  a  momentary  stop;  and  then  actually  reverse  their 
motion,  and  run  back  upon  their  former  course,  with  a  rapidity  at  first 
increasing,  then  diminishing,  till  the  reversed  or  retrograde  motion  ceases 
altogether.  Another  station,  or  moment  of  apparent  rest  or  indecision, 
now  takes  place;  after  which  the  movement  is  again  reversed,  and 
resumes  it?  original  direct  character.  On  the  whole,  however,  the  amount 
of  direct  motion  more  than  compensates  the  retrograde;  and  by  the 

Fig.  63. 


■IV 


excess  of  the  former  over  the  latter,  the  gradual  advance  of  the  planet 
from  west  to  east  is  maintained.  Thus,  supposing  the  Zodiac  to  bo 
unfolded  into  a  plane  surface,  (or  represented  as  in  Mercator's  projection, 


1    "i 


I     Itr, 

!.     fciff 


I- 


It 


liM 


fi 


244 


OUTLINES   OF   ASTRONOMY. 


art.  283,  taking  the  ecliptic  E  C  for  its  ground  line,)  the  track  of  a  planet 
when  mapped  down  by  observation  from  day  to  day,  will  offer  the  appear- 
ance P  Q  R  S,  &o. ;  the  motion  from  P  to  Q  being  direct, 'at  Q  stationary, 
from  Q  to  R  retrograde,  at  R  again  stationary,  from  R  to  S  direct,  and 
so  on. 

(460.)  In  the  midst  of  the  irregularity  and  fluctuation  of  this  motion, 
one  rcr  -  able  feature  of  uniformity  is  observed.  Whenever  the  planet 
crosses  roliptic,  as  at  N  in  the  figure,  it  is  said  (like  the  moon)  to  be 
in  its  node ;  and  as  the  earth  necessarily  lies  in  the  plane  of  the  ecliptic, 
the  planet  cannot  be  ap^^arently  or  uranograpliically  situated  in  tie 
celestial  circle  so  called,  without  being  really  and  locally  situated  in  that 
plane.  The  visible  passage  of  a  planet  through  its  node,  then,  is  a  phe- 
nomenon indicative  of  a  circumstance  in  its  real  motion  quite  independent 
of  the  station  from  which  we  view  it.  Now,  it  is  easy  to  ascertain,  by 
observation,  when  a  planet  passes  from  the  north  to  the  south  side  of  the 
ecliptic :  we  have  only  to  convert  its  right  ascensions  and  declinations  into 
longitudes  and  latitudes,  and  the  change  from  north  to  south  latitude  c  n 
two  successive  days  will  advertise  us  on  what  day  the  transition  took 
place ;  while  a  simple  proportion,  grounded  on  the  observed  state  of  its 
motion  in  latitude  in  the  interval,  will  suffice  to  fix  the  precise  hour  ?ud 
minute  of  its  arrival  on  the  ecliptic.  Now,  this  being  done  for  se\.,.  1 
transitions  from  side  to  side  of  the  ecliptic,  and  their  dates  thereby  fixed, 
we  find,  universally,  that  the  interval  of  time  elapsing  between  the  suc- 
cessive passages  of  each  planet  through  the  same  node  (whether  it  be  the 
ascending  or  the  descending)  is  always  alike,  whether  the  planet  at  the 
moment  of  such  passage  be  direct  or  retrograde,  swift  or  slow,  in  its 
apparent  movement. 

(461.)  Here,  then,  we  have  a  circumstance  which,  while  it  shows  that 
the  motions  of  the  planets  are  in  fact  subject  to  certain  laws  and  fixed 
periods;  may  lead  us  very  naturally  to  suspect  that  the  apparent  irregu- 
larities and  complexities  of  their  movements  may  be  owing  to  our  not 
seeing  them  from  their  natural  centre  (art.  338,  371),  and  from  our  mixing 
up  with  tl  cir  own  proper  motions  movements  of  a  parallactic  kind,  due  to 
our  own  change  of  place,  in  virtue  of  the  orbitual  motion  of  the  earth 
about  the  sun. 

(462.)  If  we  abandon  the  earth  as  a  centre  of  the  planetary  motions,  it 
cannot  admit  of  a  moment's  hesitation  where  we  should  place  that  centre 
with  the  greatest  probability  of  truth.  It  must  surely  be  the  sun  which 
is  entitled  to  the  first  trial,  as  a  station  to  which  to  refer  to  them.  If  it 
be  not  connected  with  them  by  any  physical  relation,  it  at  least  possesses 
the  advantage,  which  the  earth  does  not,  of  comparative  immobility.    But 


m  co^irj. 


THE   SUN  THE  CENTRE  OF   OUR   SYSTEM. 


245 


after  what  has  been  shown  in  art.  449,  of  the  immense  mass  of  that  lumi- 
nary, and  of  the  office  it  performs  to  us  as  a  quiescent  centre  of  our  orbi- 
tual  motion,  nothing  can  be  more  natural  than  to  suppose  it  may  perform 
the  same  to  other  globes  which,  like  the  earth,  may  be  revolving  round  it; 
and  these  globes  may  bo  visible  to  us  by  its  light  reflected  from  them,  as 
the  moon  is.  Now  there  are  many  facts  which  give  a  strong  support  to 
the  idea  that  the  planets  are  in  this  predicament. 

(463.)  In  the  first  place,  the  planets  really  are  great  globes,  of  a  size 
commensurate  with  the  earth,  and  several  of  them  much  greater.  When 
examined  through  powerful  telescopes,  they  are  seen  to  be  round  bodies, 
of  sensible  and  even  of  considerable  apparent  diameter,  and  ofifering  dis- 
tinct and  characteristic  peculiarities,  which  show  them  to  be  solid  masses, 
each  possessing  its  individual  structure  and  mechanism ;  and  that,  in  one 
instance  at  least,  an  exceedingly  artificial  and  complex  one.  (See  the 
representations  of  Mars,  Jupiter,  and  Saturn,  in  Plate  III.)  That  their 
distances  from  us  are  great,  much  greater  than  that  of  the  moon,  and  some 
of  them  even  greater  than  that  of  the  sun,  we  infer,  1st,  from  their  being 
occulted  by  the  moon,  and  2dly,  from  the  smallness  of  their  diurnal 
parallax,  which,  even  for  the  nearest  of  them,  when  most  favourably 
situated,  does  not  exceed  a  few  seconds,  and  for  the  remote  ones  is  almost 
imperceptible.  From  the  comparison  of  the  diurnal  parallax  of  a  celestial 
body,  with  its  apparent  semidiameter,  we  can  at  once  estimate  its  real  size. 
For  the  parallax  is,  in  fact,  nothing  else  than  the  apparent  semidiameter 
of  the  earth  as  seen  from  the  body  in  question  (art.  339  et  seq.) ;  and,  the 
intervening  distance  being  the  same,  the  real  diameters  must  be  to  each 
other  in  the  proportion  of  the  apparent  ones.  Without  going  into  parti- 
culars, it  will  suffice  to  state  it  as  a  general  result  of  that  comparison,  that 
the  planets  are  all  of  them  incomparably  smaller  than  the  sun,  but  some 
of  them  as  large  as  the  earth,  and  others  much  greater. 

(464.)  The  next  fact  respecting  them  is,  that  their  distances  from  us, 
as  estimated  from  the  measurement  of  their  angular  diameters,  are  in  a 
continual  state  of  change,  periodically  increasing  and  decreasing  within 
certain  limits,  but  by  no  means  corresponding  with  the  supposition  of 
regular  circular  or  elliptic  orbits  described  by  them  about  the  earth  as  a 
centre  oi  focus,  but  maintaining  a  constant  and  obvious  relation  to  their 
apparent  angular  distances  or  elongations  from  the  sun.  For  example ; 
the  apparent  diameter  of  Mars  is  greatest  when  in  opposition  (as  it  is 
called)  to  the  sun,  %.  e.  when  in  the  opposite  part  of  the  ecliptic,  or  when 
it  comes  on  the  meridian  at  midnight,  —  being  then  about  18",  —  but 
diminishes  rapidly  from  I'hat  to  about  4",  which  is  its  apparent  diameter 
when  in  conmnctwn-,  or  when  seen  in  nearly  the  same  direction  as  that 


't 


■.'^h 


246 


r    <    OUTLINES  OF  ASTRONOMY. 


/ 


luminary.  This,  and  facts  of  a  similar  character,  observed  with  respect 
to  the  apparent  diameters  of  the  other  planets,  clearly  point  oat  the  buq 
as  having  more  than  an  accidental  relation  to  their  movements. '" ' ' 

(466.)  Lastly,  certain  of  the  plnnets,  (Mercury,  Venus,  and  Mars,) 
when  viewed  through  telescopes,  exhibit  the  appearance  of  phases  like 
thooe  of  the  moon.  This  proves  that  they  are  opaque  bodies,  shining 
only  by  reflected  light,  which  can  be  no  other  than  that  of  the  sun'a  ] 
not  only  because  there  is  no  other  source  of  light  external  to  them  suffi- 
ciently powerful,  but  because  the  appearance  and  succession  of  the  phases 
themselves  are  (like  their  visible  diameters)  intimately  connected  with 
their  elongations  from  the  sun,  as  will  presently  be  shown. 

(466.)  Accordingly  it  is  found,  that,  when  we  refer  the  planetary  move- 
ments to  the  sun  as  a  centre,  all  that  apparent  irregularity  which  they 
offer  when  viewed  from  the  earth  disappears  at  once,  and  resolves  itself 
into  one  simple  and  general  law,  of  which  the  earth's  motion,  as  ex- 
plained in  a  former  chapter,  is  only  a  particular  case.  In  order  to  show 
how  this  happens,  let  us  take  the  case  of  a  single  planet,  which  we  will 
suppose  to  revolve  round  the  sun,  in  a  plane  nearly,  but  not  quite,  coin- 
cident with  the  ecliptic,  but  passing  through  the  sun,  and  of  course  inter- 
secting the  ecliptic  in  a  fixed  line,  which  is  the  line  of  the  planet's  nodes. 
This  line  must  of  course  divide  its  orbit  into  two  segments;  and  it  is 
evident  that,  so  long  as  the  circumstances  of  the  planet's  motion  remain 
otherwise  unchanged,  the  times  of  describi&g  these  segments  must  remain 
the  same.  The  interval,  then,  between  the  planet's  quitting  either  node, 
and  returning  to  the  same  node  again,  must  be  that  in  which  it  describes 
one  complete  revolution  round  the  sun,  or  its  periodic  time ;  and  thus  we 
are  furnished  with  a  direct  method  of  ascertaining  the  periodic  time  of 
each  planet. 

(467.)  We  have  said  (art.  457)  that  the  planets  make  the  entire  tour 
of  the  heavens  under  very  different  circumstances.  This  must  be  ex- 
plained. Two  of  them  —  Mercury  and  Venus  —  perform  this  circuit 
evidently  as  attendants  upon  the  sun,  from  whose  vicinity  they  never 
depart  beyond  a  certain  limit.  They  are  seen  sometimes  to  the  east, 
sometimes  to  the  west  of  it.  In  the  former  case  they  appear  conspicuous 
over  the  western  horizon,  just  after  sunset,  and  are  called  evening  stars : 
Venus,  especially,  appears  occasionally  in  this  situation  with  a  dazzling 
lustre ;  and  in  favourable  circumstances  may  be  observed  to  cast  a  pretty 
strong  shadow.'    When  they  happen  to  be  to  the  west  of  the  sun,  they 

'  It  must  be  thrown  upon  a  white  ground.  An  open  window  in  a  wbite-washed  room 
is  the  best  exposure.  In  this  situation  I  have  observed  not  only  the  shadow,  but  the 
diffracted  fringes  edging  its  outline.  —  H.  Note  to  the  edition  qf  1833.  Venus  majr 
often  be  seen  with  the  naked  eye  in  the  daytime. 


INFERIOR   PLANETS. 


247 


rise  before  that  luiainary  in  the  morning,  and  appear  over  the  eastern 
horizon  as  morning  stars :  they  do  not,  however,  attain  the  same  elonga- 
Hon  from  the  sun.  Mercury  never  attains  a  greater  angular  distance  from 
it  than  about  29°,  while  Venus  extends  her  excursions  on  either  side  to 
about  47**.  When  they  have  receded  from  the  sun,  eastward,  to  their 
respective  distances,  they  remain  for  a  time,  as  it  were,  immoveable  with 
reject  to  it,  and  are  carried  along  with  it  in  the  ecliptic  with  a  motion 
equal  to  its  own ;  but  presently  they  begin  to  approach  it,  or,  which  comes 
to  the  same,  their  motion  in  longitude  diminishes,  and  the  sun  gains  upon 
them.  As  this  approach  goes  on,  their  continuance  above  the  horizon 
after  sunset  becomes  daily  shorter,  till  at  length  they  set  before  the  dark- 
ness has  become  sufficient  to  allow  of  their  being  seen.  For  a  time,  then, 
they  are  not  seen  at  all,  unless  on  very  rare  occasions,  when  they  are  to 
be  observed  passing  across  the  sun's  disc  as  small,  round,  well-dejined 
Hack  spots,  totally  different  in  appearance  from  the  solar  spots  (art.  386.) 
These  phenomena  are  emphatically  called  transits  of  the  respective 
planets  across  the  sun,  and  take  place  when  the  earth  happens  to  be 
passing  the  line  of  their  nodes  while  they  are  in  that  part  of  their  orbits, 
just  as  in  the  account  we  have  given  (art.  412)  of  a  solar  eclipse.  After 
having  thus  continued  invisible  for  a  time,  however,  they  begin  to  appear 
on  the  other  side  of  the  sun,  at  first  showing  themselves  only  for  a  few 
minutes  before  sunrise,  and  gradually  longer  and  longer  as  they  recede 
from  him.  At  this  time  their  motion  in  longitude  is  rapidly  retrograde. 
Before  they  attain  their  greatest  elongation,  however,  they  become  station- 
ary in  the  heavens  j  but  their  recess  from  the  sun  is  still  maintained  by 
the  advance  of  that  luminary  along  the  ecliptic,  which  continues  to  leave 
them  behind,  until,  having  reversed  their  motion,  and  become  again 
direct,  they  acquire  sufficient  speed  to  commence  overtaking  him  —  at 
which  moment  they  have  their  greatest  western  elongation :  and  thus  is  a 
kind  of  oscillatory  movement  kept  up,  while  the  general  advance  along 
the  ecliptic  goes  on. 

(468.)  Suppose  P  Q  to  be  the  ecliptic,  and  A  B  D  the  orbit  of  one  of 
these  planets,  (for  instance,  Mercury,)  seen  almost  edgewise  by  an  eye 
situated  very  nearly  in  its  plane  ;  S,  the  sun,  its  centre;  and  A,  B,  D,  S, 
successive  positions  of  the  planet,  of  which  B  and  S  are  in  the  nodes. 
If,  then,  the  sun  S  stood  apparently  still  in  the  ecliptic,  the  planets  would 
simply  appear  to  oscillate  backwards  and  forwards  from  A  to  D,  alter- 
nately passing  before  and  behind  the  sun ;  and,  if  the  eye  happened  to 
lie  exactly  in  the  plane  of  the  orbit,  transiting  his  disc  in  the  former 
case,  and  being  covered  by  it  in  the  latter.  But  as  the  sun  is  not  so 
stationary,  but  apparently  carried  along  the  ecliptic  P  Q,  let  it  be  supposed 


24S 


OUTLINES   OF  ASTRONOMY. 


to  move  over  the  spaces  S  T,  T  U,  U  V,  while  the  planet  in  each  case  exe- 
cutes one  quarter  of  its  period.  Then  wil>  its  orbit  be  apparently  caiTied 
along  with  the  sun,  into  the  successive  positions  represented  in  the  figure 
and  while  its  real  motion  round  the  sun  brings  it  into  the  respective  points 
B,  D,  S,  A,  its  apparent  movement  in  the  heavens  will  seem  to  have  been 
along  the  wavy  or  zigzag  line  A  N  H  K.  In  this,  its  motion  in  longitude 
will  have  been  direct  in  the  parts  A  N,  N  H,  and  retrograde  in  the  parts 
H  71 K ;  while  at  the  turns  of  the  zigzag,  as  at  H,  it  will  have  been  sta- 
tionary. 

(4G9.)  The  only  two  planets  —  Mercury  and  Venus  —  whose  evolu- 
tions are  such  as  above  described,  are  called  inferior  planets ;  their  points 
of  farthest  recess  from  the  sun  are  called  (as  above)  their  greatest  eastern 
and  western  elongations  ;  and  their  points  of  nearest  approach  to  it,  their 
inferior  and  superior  conjunctions, — the  former  when  the  planet  passes 
between  the  earth  and  the  sun,  the  latter  when  behind  the  sun. 

(470.)  In  art.  467  we  have  traced  the  apparent  path  of  an  inferior 
planet,  by  considering  its  orbit  in  section,  or  as  viewed  from  a  point  in 
the  plane  of  the  ecliptic.  Let  us  now  contemplate  it  in  plan,  or  as  viewed 
from  a  station  above  that  plane,  and  projected  on  it.  Suppose  then,  S  to 
represent  the  sun,  abed  the  orbit  of  Mercury,  and  A B  C D  a  part  of 
that  of  the  earth  —  the  direction  of  the  circulation  being  the  same  in 

Fig.  65.  •         • 


;/ 


both,  viz,  that  of  the  arrow.  When  the  planet  stands  at  a,  let  the  earth 
be  si^juated  at  A,  in  the  direction  of  a  tangent,  a  A,  to  its  orbit ;  then  it 
is  eviaent  that  it  will  appear  at  its  greatest  elongation  from  the  sun,— 


INFERIOR   PLANETS. 


240 


the  angle  a  A  S,  which  measures  their  apparent  interval  as  seen  from  A, 
being  then  greater  than  in  any  other  situation  of  a  upon  its  own  circle. 

(471.)  Now,  this  angle  being  known  by  observation,  we  are  hereby 
furnished  with  a  ready  means  of  ascertaining,  at  least  approximately,  tbo 
distance  of  the  |)Ianet  from  the  sun,  or  the  radius  of  its  orbit,  supposed  a 
circle.  For  the  triangle  S  A  a  is  right-angled  at  a,  and  consequently  we 
have  S  a :  S  A  :  :  sin.  S  A  a :  radius,  by  which  proportion  the  radii  S  a, 
S  A  of  the  two  orbits  are  directly  compared.  If  the  orbits  wore  both 
exact  circles,  this  would  of  course  be  a  perfectly  rigorous  mode  of  proceed- 
ing: but  (as  is  proved  by  the  inequal'ty  of  the  resulting  values  of  S  a 
obtained  at  different  times)  this  is  not  the  case }  and  it  becomes  neces«<ary 
to  admit  an  ezcentricity  of  position,  and  a  deviation  from  the  exact  circu- 
lar form  in  both  orbits,  to  account  for  this  difference.  Neglecting,  how- 
ever, at  present  this  inequality,  a  mean  or  average  value  of  S  a  may,  at 
least,  be  obtained  from  the  frequent  repetition  of  this  process  in  all  vari- 
eties of  situation  of  the  two  bo'lies.  The  calculations  being  performed, 
it  is  concluded  that  the  mean  distance  of  Mercury  from  the  sun  is 
about  36000000  miles;  and  that  of  Venus,  similarly  derived,  about 
68000000;  the  radius  of  the  earth's  orbit  being  95000000. 

(472.)  The  sidereal  periods  of  the  planets  may  bo  obtained  (as  before 
observed),  with  a  considerable  approach  to  accuracy,  by  observing  their 
passages  through  the  nodes  of  their  orbits;  and  indeed,  when  a  certain 
very  minute  motion  of  these  nodes  and  the  apsides  of  their  orbits  (similar 
to  that  of  the  moon's  nodes  and  apsides,  but  incomparably  slower)  is 
allowed  for,  with  a  precision  only  limited  by  the  imperfection  of  the 
appropriate  observations.  By  such  observations,  so  corrected,  it  appears 
that  the  sidereal  period  of  Mercury  is  87*  23"  IS"  43 -O*;  and  that  of 
Venus,  224*  16"  49"  80'.  These  periods,  however,  are  widely  different 
from  the  intervals  at  which  the  successive  appearances  of  the  two  planets 
at  their  eastern  and  western  elongations  from  the  sun  are  observed  to 
happen.  Mercury  is  seen  at  its  greatest  splendour  as  an  evening  star,  at 
average  intervals  of  about  116,  and  Venus  at  intervals  of  about  584  days. 
The  difference  betwcon  the  sidereal  and  si/nodical  revolutions  (art.  418) 
accounts  for  this.  Keferring  again  to  the  figure  of  art.  470,  if  the  sun 
stood  still  at  A,  while  the  planet  advanced  in  its  orbit,  the  lapse  of  a 
sidereal  period,  which  should  bring  it  round  again  to  a,  would  also  produce 
a  similar  elongation  from  the  sun.  But,  meanwhile,  the  earth  has 
advanced  in  its  orbit  in  the  same  direction  towards  E,  and  therefore  the 
next  greatest  elongation  on  the  same  side  of  the  sun  will  happen  —  not 
in  the  position  a  A  of  the  two  bodies,  but  in  some  more  advanced  posi- 
tion, e  E.     The  determination  of  this  position  depends  on  a  calculation 


.  t  li'ii 


'  U  V'l 


t  '^'"X 


260 


OUTLINES  OP  ABTRONOMT. 


i 


exactly  similar  to  what  has  beoo  explained  in  the  artiole  referred  to ;  and 
wo  need,  tbtrefore,  only  state  the  resulting  synodical  revolutions  of  the 
two  planets,  which  come  out  respectively  116'877*,  and  688-920*.     ■ 

(473.)  In  this  interval,  the  planet  will  have  described  a  whole  revolu- 
tion j)lu8  the  are  acCf  and  the  earth  only  the  are  ACE  of  its  orbit. 
During  its  lapse,  the  inferior  conjunction  will  happen  when  the  earth  has 
a  certain  intermediate  situation,  B,  and  the  planet  has  reoohed  b,  a  point 
between  the  sun  and  earth.  The  greatest  elongation  on  the  oppoaite  side 
of  the  sun  will  happen  when  the  earth  has  come  to  C,  and  the  planet  to 
c,  where  the  line  of  junction  0  c  is  a  tangent  to  the  interior  circle  on  the 
opposite  side  from  M.  Lastly,  the  superior  conjunction  will  happen 
when  the  earth  arrives  at  D,  and  the  planet  at  d  in  the  same  line  pro- 
longed on  the  other  side  of  the  sun.  The  intervals  at  which  those  phe- 
nomena happen  may  easily  be  computed  from  a  knowledge  of  the  synodi- 
cal periods  and  the  radii  of  the  orbits. 

(474.)  The  ciroumforences  of  circles  are  in  the  proportion  of  their 
radii.  If,  then,  we  calculate  the  circumferences  of  the  orbits  of  McrcUrj 
and  Venus,  and  the  earth,  and  compare  them  with  the  times  in  which 
their  revolutions  are  performed,  we  shall  find  that  the  actual  velocities 
with  which  they  move  in  their  orbits  diflFer  greatly;  that  of  INIercury 
being  about  109360  miles  per  hour,  of  Venus  80000,  and  of  the  earth 
G8040.  From  this  it  follows,  that  at  the  inferior  conjunction,  or  at  b, 
either  planet  is  moving  in  the  same  direction  as  the  earth,  but  with  a 
greater  velocity;  it  will,  therefore,  leave  the  earth  behind  it;  and  the 
apparent  motion  of  the  planet  viewed  from  the  earth,  will  be  as  if  the 
planet  stood  still,  and  the  earth  moved  in  a  contrary  direction  from  what 
it  really  does.  In  this  situation,  then,  the  apparent  motion  of  the  planet 
must  be  contrary  to  the  apparent  motion  of  the  sun;  and,  therefore, 
retrograde.  On  the  other  hand,  at  the  superior  conjunction,  the  real 
motion  of  the  planet  being  in  the  opposite  direction  to  that  of  the  earth, 
the  relative  motion  will  be  the  same  as  if  the  planet  stood  still,  and  the 
earth  advanced  with  their  united  velocities  in  its  own  proper  direction. 
In  this  situation,  then,  the  apparent  motion  will  be  direct.  Both  these 
results  are  in  accordance  with  observed  fact. 

(475.)  The  stationary  points  may  be  determined  by  the  following  con- 
sideration. At  a  or  c,  the  points  of  greatest  elongation,  the  motion  of 
the  planet  is  directly  to  or  from  the  earth,  or  along  their  line  of  junction, 
while  that  of  tbe  earth  is  nearly  perpendicular  to  it.  Here,  then,  the 
apparent  motion  must  be  direct.  At  b,  the  inferior  conjunction,  we  have 
seen  that  it  must  be  retrograde,  owing  to  the  planet's  motion  (which  is 
there,  as  well  as  the  earth's,  perpendicular  to  tbe  line  of  junction)  snr- 


INFERIOR  PLANETS. 


251 


/ 


passing  tlic  earth's.  Hence,  the  stationary  points  ought  to  lie,  as  it  is 
fuuiid  by  observation  they  do,  between  a  and  b,  or  c  and  b,  viz.  in  saoh  u 
position  thai  the  obliquity  of  the  planet's  motion  with  respect  to  the  lino 
i)t'  junction  shall  just  oompensate  for  the  excess  of  its  velocity,  and  cause 
an  equal  advance  of  eaoh  extremity  of  that  line,  by  the  motion  of  the 
planet  at  one  und,  and  of  the  earth  at  the  other :  so  that,  for  an  instant 
of  time,  the  whole  line  shall  move  parallel  to  itself.  The  question  thus 
proposed  is  purely  geometrical,  and  its  solution  on  the  supposition  of 
circulur  orbits  is  easy.    Let  E  e  and  P  j>-  represent  small  arcs  of  the 

Fig.  66.  , 


■(  y^!" 


/' 

\  > 

'\  i"" 

j.-ij 

a' 

1 

1 

J     ; 

,>, 

r 

.?■ 

at 

1 

>i^ 

"i, 

orbits  of  the  earth  and  planet  described  contemporaneously,  at  the  moment 
when  the  latter  appears  stationary,  about  S,  the  sun.  Produce  p  P  and 
e  E,  tangents  at  P  and  E,  to  meet  at  B,  and  prolong  E  P  backwards  to 
Q,  join  ep.  Then  since  P  E,  jj  e  are  parallel,  in«  hive  by  similar  triali- 
gles  Pjj:Ec::PR:RE,  and  since,  putting  v  .ad  V  for  the  respec- 
tive velocities  of  the  planet  and  the  earth,  Pp  :  E  e  : :  v  :  V ;  therefore 


:  V  : :  P  R  :  R  E  : :  sin.  PER:  sin.  li  PR       ^ 
: :  cos.  SEP:  ccj.  3  P  Q 
•  ■ '■'  : :  cos.  SEP:  cos.  (S  E  P+E  S  P) 


because  the  angles  S  E  R  and  S  P  R  are  right  angles.     Moreover,  if  r 
and  R  be  the  radii  of  the  respective  orbits,  we  have  also     » v.  -  .,.^«^>-j,j^. 


r  :  R  : :  sin.  SEP:  sin.  (S  E  P+E  S  P)  ;t  ; 

from  which  two  relations  it  is  easy  to  deduce  the  values  of  the  two  angles 
SEP  and  E  S  Pj  the  former  of  which  is  the  apparent  elongation  of  the 


1      fl 


252 


OUTLINES  OF  ASTRONOMY. 


])laQct  from  the  sud,'  the  latter  the  difference  of  belioceuirio  loogitudes 
of  the  earth  and  planet.         >,  -  ,-^    .  ';      •     ♦     5  '    ' 

(476.)  When  we  regard  the  orbits  as  other  than  circles  (which  they 
really  are),  the  problem  becomes  somewhat  complex  —  too  much  so  to  bo 
here  entered  upon.  It  will  suffice  to  state  the  results  which  experience 
verifies,  and  which  assigns  the  stationary  points  of  Mercury  at  from  15° 
to  20°  of  elongation  from  the  sun,  according  to  circumstances;  and  of 
Venus,  at  an  elongation  never  varying  much  from  29°.  The  former  con- 
tinues to  retrograde  during  about  22  days ;  the  latter,  about  42. 

(477.)  We  have  said  that  some  of  the  planets  exhibit  phases  like  the 
moon.  This  is  the  case  with  both  Mercury  and  Venus ;  and  is  readily 
explained  by  a  consideration  of  their  orbits,  such  as  wo  have  above  sup- 
posed them.  In  fact,  it  requires  little  more  than  mere  inspection  of  the 
figure  annexed,  to  show,  that  to  a  spectator  situated  on  the  earth  E,  an 


inferior  planet,  illuminated  by  the  sun,  and  therefore  bright  on  the  side 
next  to  him,  and  dark  on  that  turned  from  him,  will  appear /t<?Z  at  the 
superior  conjunction  A ;  gihhoxis  {i.  e.  more  than  half  full,  like  the  moon 
between  the  first  and  second  quarter)  between  that  point  and  the  points 
B  C  of  its  greatest  elongation ;  half-mooned  at  these  points ;  and  crescent- 
shaped,  or  horned,  between  these  and  the  inferior  conjunction  D.  As  it 
approaches  this  point,  the  crescent  ought  to  thin  off  till  it  vanishes  alto- 
gether, rendering  the  planet  invisible,  unless  in  those  cases  where  it 
transits  the  sun's  disc,  and  appears  on  it  as  a  black  spot.  All  these  phe- 
nomena are  exactly  conformable  to  observation. 

(478.)  The  variation  in  brightness  of  Venus  in  different  parts  of  its 
apparent  orbit  is  very  remarkable.     This  arises  from  two  causes :  1st,  the 

R  V 

'If — =»m  and — =n,  SEP=^,  ESP=i//,  the  equations  to  be  resolved  are  ii'n. 


i^+i//)— m  «n.  f,  and  co$.  (^+4)=m  cot.  </»,  which  give  cos.  \p= 


1+mn 
tn+n 


TRANSITS  OP  VENUS  AND   MERCURY. 


253 


varying  proportion  of  its  visible  illuminated  area  to  its  whole  disc ;  and, 
2dly,  the  varying  angular  diameter,  or  whole  apparent  magnitude  of  the 
disc  itself.  As  it  approaches  its  inferior  conjunction  from  its  greater 
elongation,  the  half-moon  becomes  a  crescent,  which  thtna  of;  but  this  is 
more  than  compensated,  for  some  time,  by  the  increasing  apparent  magni- 
tude, in  consequence  of  its  diminishing  distance.  Thus  the  total  light 
received  from  it  goes  on  increasing,  till  at  length  it  attains  a  maximum, 
which  takes  place  when  the  planet's  elongation  is  about  40". 

(479.)  The  transits  of  Venus  are  of  very  rare  occurrence,  taking  place 
alternately  at  the  very  unequal  but  regularly  recurring  intervals  of  8, 122, 
8, 105,  8,  122,  &c.,  years  in  succession,  and  always  in  June  or  December. 
As  astronomical  pho3nomena,  they  are  extremely  important ;  since  they 
afford  the  best  and  most  exact  means  we  possess  of  ascertaining  the  sun's 
distance,  or  its  parallax.  Without  going  into  the  niceties  of  calculation 
of  this  problem,  which,  owing  to  the  great  multitude  of  circumstances  to 
be  attended  to,  are  extremely  intricate,  we  shall  here  explain  its  principle, 
which,  in  the  abstract,  is  very  simple  and  obvious.  Let  E  be  the  earth, 
V  Venus,  and  S  the  sun,  and  C  D  the  portion  of  Venus's  relative  orbit 
which  she  describes  while  in  the  act  of  transiting  the  sun's  disc.  Suppose 
A  B  two  spectators  at  opposite  extremities  of  that  diameter  of  the  earth 

Fig.  67. 


which  is  perpendicular  to  the  ecliptic,  and,  to  avoid  complicating  the  case, 
let  us  lay  out  of  consideration  the  earth's  rotation,  and  suppose  A,  B,  to 
retain  that  situation  during  the  whole  time  of  the  transit.  Then,  at  any 
moment  when  the  spectator  at  A  sees  the  centre  pf  Venus  projected  at  a 
on  the  sun's  disc,  he  at  B  will  see  it  projected  at  b.  If  then  one  or  other 
spectator  could  suddenly  transport  himself  from  A  to  B,  he  would  see 
Venus  suddenly  displaced  on  the  disc  from  a  to  6 ;  and  if  he  had  any 
means  of  noting  accurately  the  place  of  the  points  on  the  disc,  either  by 
micrometrical  measures  from  its  edge,  or  by  other  means,  he  might  ascer- 
tain the  angular  measure  of  a  b  as  seen  from  the  earth.  Now,  since 
A  V  or,  B  V  6,  are  straight  lines,  and  therefore  make  equal  angles  on  each 
side  V,  a  6  will  be  to  A  B  as  the  distance  of  Venus  from  the  sun  is  to  its 
distance  from  the  earth,  or  as  68  to  27,  or  nearly  as  2}  to  1  j  ab  there- 


in 


i  •: 

i       I 


i\ 


iikA 


254 


OUTLINES  OF  ASTRONOMY. 


fore  occupies  on  the  sun's  disc  a  space  2^  times  as  great  as  the  earth's 
diameter;  and  its  angular  measure  is  therefore  equal  to  about  2^  times 
the  earth's  apparent  diameter  at  the  distance  of  the  sun,  or  (which  h  the 
same  thing)  to  five  times  the  sun's  horizontal  parallax  (art.  339).  Any 
error,  therefore,  which  may  be  committed  in  measuring  a  b,  will  entail 
only  (me  fifth  of  that  error  on  the  horizontal  parallax  concluded  from  it. 

(480.)  The  thing  to  be  ascertained,  therefore,  is,  in  fact,  neither  more 
nor  less  than  the  breadth  of  the  zone  P  Q  R  S,  j)  g  r«,  included  between 
the  extreme  apparent  paths  of  the  centre  of  Yenus  across  the  sun's  disc, 
from  its  entry  on  one  side  to  its  quitting  it  on  the  other.  The  whole 
business  of  the  observers  at  A,  B,  therefore,  resolves  itself  into  this  j — to 
ascertain,  with  all  possible  care  and  precision,  each  at  his  own  station,  this 
path, — ^where  it  enters,  where  it  quits,  and  what  segment  of  the  sun's  disc 
it  cuts  off.  Now,  one  of  the  most  exact  ways  in  which  (conjoined  with 
careful  micrometric  measures)  this  can  be  done,  is  by  noting  the  time  occu- 
pied  in  the  whole  transit ;  for  the  relative  angular  motion  of  Venus  being, 
in  fact,  very  precisely  known  from  the  tabks  of  her  motion,  and  the  appt- 
rent  path  being  very  nearly  a  straight  line,  these  times  give  us  a  measure 
(on  a  very  enlarged  scale)  of  the  lengths  of  the  chords  of  the  segments 
cut  off;  and  the  sun's  diameter  being  known  also  with  great  precision, 
their  versed  sines,  and  therefore  their  difference,  or  the  breadth  of  the 
zone  required,  becomes  known.  To  obtain  these  times  correctly,  each 
observer  must  ascertain  the  instants  of  ingress  and  egress  of  the  centre. 
To  do  this,  he  must  note,  1st,  the  instant  when  the  first  visible  impression 
or  notch  on  the  edge  of  the  disc  at  P  is  produced,  or  the  Jirst  external 
contact;  2dly,  when  the  planet  is  just  wholly  immersed,  and  the  broken 
edge  of  the  dise  just  closes  again  at  Q,  or  the  first  internal  contact;  and, 
lastly,  he  must  make  the  same  observations  at  the  egress  at  11,  S.  The 
mean  of  the  internal  end  external  contacts,  corrected  for  the  curvature  of 
the  sun's  limb  in  the  intervals  of  the  respective  points  of  contract,  internal 
and  external,  gives  the  entry  and  egress  of  the  planet's  centre. 

(481.)  The  modifications  introduced  into  this  process  by  the  earth's 
rotation  on  ii  axis,  and  by  other  geographical  stations  of  the  observers 
thereon  than  here  supposed,  are  similar  in  their  principles  to  those  which 
enter  into  the  calculation  of  a  solar  eclipse,  or  the  occultation  of  a  star  by 
the  moon,  only  more  refined.  Any  consideration  of  them,  however,  here, 
would  lead  us  too  far;  but  in  the  view  we  have  tii]:eii  of  the  subject,  it 
affords  an  admirable  example  of  the  way  in  whit  h  uiiuute  elements  in 
astronomy  may  become  maguified  in  their  effects,  and,  by  being  made  sub- 
ject to  measurement  on  a  greatly  enlarged  scale,  or  by  substituting  the 
measure  of  time  for  space,  may  be  ascertained  with  a  degi'ce  of  precision 


SUPERIOR  PLANETS. 


:oo 


adequate  to  every  purpose,  by  only  watching  favourable  opportunities,  and 
taking  advantage  of  nicely  adjusted  combinations  of  circumstance.  So  im- 
portant has  this  observation  appeared  to  astronomers,  that  at  the  last  transit 
of  Venus,  in  1769,  expeditions  were  fitted  out,  on  the  most  extensive  scale, 
by  the  British,  French,  Russian,  and  other  governments,  to  the  remotest 
corners  of  the  globe,  for  the  express  purpose  of  performing  it.  The  cele- 
brated expedition  of  Captain  Cook  to  Otaheite  was  one  of  them.  The  gene- 
ral result  of  all  the  observations  made  on  this  most  memorable  occasion  gives 
8"5776  for  the  sun's  horizontal  parallax.  The  two  next  occurrences  of 
this  phaenomenon  will  happen  on  Deo.  8,  1874,  and  Deo.  6,  1882. 

(482.)  The  orbit  of  Mercury  is  very  elliptical,  the  excentricity  being 
nearly  one  fourth  of  the  mean  distance.  This  appears  from  the  inequality 
of  the  greatest  elongations  from  the  sun,  as  observed  at  different  times, 
and  which  vary  between  the  limits  16°  12'  and  28°  48',  and,  from  exact 
measures  of  such  elongations,  it  is  not  diflBcult  to  show  that  the  orbit  of 
Venus  also  is  slightly  excentric,  and  that  both  these  planets,  in  fact, 
describe  ellipses,  having  the  sun  in  their  common  focus. 

(483.)  Transits  of  Mercury  over  the  sun's  disc  occasionally  occur,  as 
in  the  case  of  Venus,  but  more  frequently ;  those  at  the  ascending  node 
in  November,  at  the  descending  in  May.  The  intervals  (considering 
each  node  separately)  are  usualli/  either  13  or  7  years,  and  in  the  order 
13, 13,  13,  7,  &c. ;  but  owing  to  the  considerable  inclination  of  the  orbit 
of  Mercury  to  the  ecliptic,  this  cannot  be  taken  as  an  exact  expression 
of  the  said  recurrence,  and  it  requires  a  period  of  at  least  217  years  to 
bring  round  the  transits  in  regular  order.  One  will  occur  in  the  present 
year  (1848,)  the  next  in  1861.  They  are  of  much  less  astronomical 
importance  than  that  of  Venus,  on  account  of  the  proximity  of  Mercury 
to  the  sun,  which  affords  a  much  less  favourable  combination  for  the 
determination  of  the  sun's  parallax. 

(484.)  Let  us  now  consider  the  superior  planets,  or  tRose  whose  orbits 
enclose  on  all  sides  that  of  the  earth.  That  they  do  so  is  proved  by 
several  circumstances:  —  1st,  They  arc  not,  like  the  inferior  planets,  con- 
fined to  certain  limits  of  elongation  from  the  sun,  but  appear  at  all  dis- 
tances from  it,  even  in  the  opposite  quarter  of  the  heavens,  or,  as  it  is 
called,  in  opposition ;  which  could  not  happen,  did  not  the  earth  at  such 
tiu;9s  place  itself  between  them  and  the  sun :  2dly,  They  never  appear 
horned,  like  Venus  or  Mercury,  nor  even  semilunar.  Those,  on  the 
contrary,  which,  from  the  minuteness  of  their  parallax,  we  conclude  to  be 
the  most  distant  from  us,  viz.  Jupiter,  Saturn,  Uranus,  and  Neptune, 
never  appear  otherwise  than  round ;  a  sufficient  proof,  of  itself,  that  we 
see  them  always  in  a  direction  not  very  remote  from  that  in  which  the 


ili 


2o6 


OUTLINES   OF  ASTRONOMY. 


i 

Si 

I 


il  ! 


sun's  rays  illuminate  them ,  and  that,  therefore,  we  occupy  a  station  which 
is  never  very  widely  removed  from  the  centre  of  their  orbits,  or,  in  other 
words,  that  the  eariili's  orbit  is  entirely  enclosed  within  theirs,  and  of 
comparatively  small  diameter.  Only  one  of  them,  Mars,  exhibits  any 
perceptible  phase,  and  in  its  deficiency  from  a  circular  outline,  never 
surpasses  a  moderate)  /  ^rtitous  appearance,  —  the  enlightened  portion  of 
the  disc  being  never  jiess  than  seven-eighths  of  the  whole.  To  understand 
this,  we  need  only  ;  ost  r-ur  eyes  on  the  annexed  figure,  in  which  E  is  the 
earth,  at  its  apparent  greatest  elongation  from  the  sun  S,  as  seen  from 
Mai's,  M.     lu  this  position,  the  angle  S  M  E,  included  between  the  Imea 

Fig.  69. 


S  M  and  E  M,  is  at  its  maximum ;  and  therefore,  in  this  state  of  things, 
a  spectator  on  the  earth  is  enabled  to  see  a  greater  portion  of  the  dark 
hemisphere  of  Mars  than  in  any  other  situation.  The  extent  of  the 
phase,  then,  or  greatest  observable  degree  of  gibbosity,  affords  a  measure 
—  a  sure,  although  a  coarse  and  rude  one  —  of  the  angle  S  M  E,  and 
therefore  of  the  proportion  of  the  distance  S  M,  of  Mars,  to  S  E,  that  of 
the  earth  f.;om  the  sun,  by  which  it  appears  that  the  diameter  of  the 
orbit  of  Mars  cannot  be  less  than  1^  times  that  of  the  earth's.  Tlie 
phases  of  Jupiter,  Saturn,  Uranus,  and  Neptune,  being  imperceptible,  it 
follows  that  their  orbits  must  include  not  only  that  of  the  earth,  but  of 
Mars  also. 

(485.)  All  the  superior  planets  are  retrograde  in  their  apparent 
motions  when  in  opposition,  and  for  some  time  before  and  after ;  but  they 
differ  greatly  from  each  other,  both  in  the  extent  of  their  arc  of  retrogra- 
dation,  in  the  duration  of  their  retrograde  movement,  and  in  its  rapidity 
when  swiftest.     It  is  more  extensive  and  rapid  in  the  case  of  Mars  than 


1-    ( 


SUPERIOR   PLANETS. 


257 


of  Jupiter,  of  Jupiter  than  of  Saturn,  of  that  planet  than  of  Uranus, 
and  of  Uranus  again  than  Neptune.  The  angular  velocity  with  which  a 
planet  appears  to  retrograde  is  easily  ascertained  by  observing  its  apparent 
place  in  the  heavens  from  day  to  day  j  and  from  such  observations,  made 
about  the  time  of  opposition,  it  is  easy  to  conclude  the  relative  magni- 
tudes of  their  orbits,  as  compared  with  the  earth's,  supposing  their 
periodical  times  known.  For,  from  these,  their  mean  angular  velocities 
are  known  also,  being  inversely  as  the  timea.    Suppose,  then,  E  e  to  be  a 


very  small  portion  of  the  earth's  orbit,  and  M  m  a  correspondiug  portion 
of  that  of  a  superior  planet,  described  on  the  day  of  opposition,  about 
the  sun  S,  on  which  day  the  three  bodies  lie  in  one  straight  line  S  E  M  X. 
Then  the  angles  E  S  e  and  M  S  m  are  given.  Now,  if  e  m  be  joined  and 
prolonged  to  meet  S  M  continued  in  X,  the  angle  e  X  E,  which  is  equal 
to  the  alternate  angle  X  e  Y,  is  evidently  the  retrogradation  of  Mars  on 
that  day,  and  is,  therefore,  also  given.  E  e,  therefore,  and  the  angle 
E  X  c,  being  given  in  the  right-angled  triangle  E  e  X,  the  side  E  X  is 
easily  calculated,  and  thus  S  X  becomes  known.  Consequently,  in  the 
triangle  S  m  X,  we  have  given  the  side  S  X  and  the  two  angles  m  S  X, 
and  m  X  S,  whence  the  other  sides,  S  m,  m  X,  are  easily  determined. 
Now,  S  m  is  no  other  than  the  radius  of  the  orbit  of  the  superior  plnact 
required,  which  in  this  calculation  is  supposed  circular,  as  well  as  that  of 
the  earth ;  a  supposition  not  exact,  but  sufficiently  so  to  afford  d  i  atisfac- 
tory  approximation  to  the  dimensions  of  its  orbit,  and  which,  .'f  the 
process  be  often  repeated,  in  every  variety  of  situation  at  which  the 
opposition  can  occur,  will  ultimately  afford  an  average  or  mean  va^ue  of 
its  diameter  fully  to  be  depended  upon. 

(486.)  To  apply  this  principle,  however,  to  practice,  it  is  necessary  to 
know  the  periodic  times  of  the  several  planets.  These  may  be  obtained 
directly,  as  has  been  already  stated,  by  observing  the  intervals  of  their 
passages  through  the  ecliptic  >  but,  owing  to  the  very  small  inclination  of 
the  orbits  of  some  of  them  to  its  plane,  they  cross  it  so  obliquely  that 
the  precise  moment  of  their  arrival  on  it  is  cot  ascertainable,  unless  by 
very  nice  observations.  A  better  method  consists  in  determining,  from 
the  observations  of  several  successive  days,  the  exact  moments  of  their 
arriving  in  opposition  with  the  sun,  the  criterion  of  which  is  a  difference 
of  longitudes  between  the  sun  and  planet  of  exactly  180**.  The  mtervai 
17 


♦'r1 


I 


-.1 


I        l!^l 


:fi 


mm 


W 


;.*  If 


r:^li;.^: 


258 


OUTLINES   OP  ASTRONOMY. 


Pi 


between  successive  oppositions  thus  obtained  is  nearly  one  synodical  no. 
riod ;  and  would  bo  exactly  so,  were  the  planet's  orbit  and  that  of  the 
earth  both  circles,  and  uniformly  described ;  but  as  that  is  found  not  to 
bo  the  c-ise  (and  tho  criterion  is,  the  incquaUti/  of  .utoosbHo  svnodk'^l 
revolutions  so  observed),  the  average  of  a  great  uurnbor,  f,;i>cji  iu  all  vn- 
rictics  of  situation  in  \;hich  tho  oppositions  occur,  wsU  L  i  freed  from  fl><, 
elliptic  inoquali'y,  and  may  bo  taken  as  a  in- an  st/ixo'u  ul  I'i'.iou  From 
this,  by  tho  considerations  and  I-y  tlio  process  of  calculation,  indicated 
(art.  418)  tho  sidereal  periods  arc  roa'lily  obtaiiiod.  'Jiio  accuracy  of  this 
detorminntion  will,  of  course,  be  grcffJly  incroaacd  by  embracing  a  lone 
interval  between  the  Ciitromo  obscivutions  oraploycd.  in  point  oi'  iuft, 
that  interval  extends  to  nearly  2000  years  in  the  cruics  of  tlio  piuiets 
known  to  tho  ancients,  who  have  recorded  their  ob3ervul.iona  *"  tlie»ii  in 
a  m;uiU(  r  sufficieiiUy  careful  to  bo  made  use  of.  Their  pi^riods  luaj-,  tliero- 
fore,  Jt  regarded  as  asccvtaintid  with  the  utmo.st  exactness.  Their  nume- 
rical Viiiuos  Tt.'il  bo  foun'J  stated,  as  well  as  tho  mean  distances,  and  all 
the  other  •.!■•  i.^.-nts  tf  the  planetary  orbits,  in  the  synoptic  table  at  tl.o 
end  of  the  voiaujo,  to  which  (to  avoid  repetition)  the  reader  is  once  for 
all  roforred. 

(487.)  Ju  casting  our  eyes  down  the  list  of  tho  ^jlanetary  distances,  and 
comparing  them  with  the  periodic  times,  we  cannot  but  bo  struck  with  a 
certain  correspondence.  The  greater  the  distance,  or  Iho  larger  the  orbit, 
evid.;ntly  the  longer  the  period.  Tho  order  of  the  planets,  beginning 
from  the  sun,  is  the  same,  whether  wo  arrange  thera  according  to  their 
distances,  or  to  the  time  they  occupy  in  completing  their  revolutions ;  and 
is  as  follows  :  —  Mercury,  Venus,  Earth,  Mars, —  tho  ultra-zodiacal  pla- 
nets, or,  as  they  are  sometimes  also  called.  Asteroids, — Jupiter,  Saturn,  Ura- 
nus, and  Neptune.  Nevertheless,  when  we  come  to  examine  the  numbers 
expressing  them,  we  find  that  tho  relation  between  the  two  series  is  not 
that  of  simple  proj;)o/-/io«anncrease.  The  periods  increase  more  than  in 
proportion  to  the  distances.  Thus,  the  period  of  Mercury  is  about  88  days, 
and  that  of  the  earth  365  —  being  in  proportion  as  1  to  4'15,  while  their 
distances  aro  in  the  less  proportion  of  1  to  2-56;  and  a  similar  remark 
holds  good  in  every  instance.  Still,  the  ratio  of  increase  of  the  times  is 
not  so  rapid  as  that  of  the  squares  of  tho  distances.  The  square  of  256 
is  6'5536,  which  is  considerably  greater  than  4"15.  Au  intermediate 
rate  of  increase,  between  the  simple  proportion  of  the  distances  and  that 
of  theii*  squares  is  therefore  clearly  pointed  out  by  the  sequence  of  tho 
numbers ;  but  it  required  no  ordinary  penetration  in  the  illustrious  Kep- 
ler, backed  by  uncommon  perseverance  and  industry,  at  a  period  when 
the  data  themselves  were   involved  in  obscurity,  and  when   the  pro 


11 
! 


I 


KEPLER  S  THIRD   LAW. 


259 


ocsst'S  of  trigonometry  and  of  numerical  calculation  were  encumbered 
with  difficulties,  of  which  the  more  recent  invention  of  logarithmic  tables 
has  happily  left  us  no  conception,  to  perceive  and  demonstrate  the  real 
law  of  their  connection.  This  connection  is  expressed  in  the  following 
proposition  :  — "  The  squares  of  the  periodic  times  of  any  two  planets  are 
to  each  other,  in  the  same  proportion  as  the  cubes  of  their  mean  distances 
from  the  sun."  Take,  for  example,  the  Earth  and  Mars,*  whose  periods 
are  in  the  proportion  of  3652564  to  6869796,  and  whose  distance  from 
the  sun  is  that  of  100000  to  152369;  and  it  will  be  found,  by  any  one 
who  will  take  the  trouble  to  go  through  the  calculation,  that — 
(3652564)' :  (6869796)" : :  (100000)» :  (152369)'. 

(488.)  Of  all  the  laws  to  which  induction  from  pure  observation  has 
ever  conducted  man,  this  third  Into  (as  it  is  called)  of  Kepler  may  justly 
be  regarded  as  the  most  remarkable,  and  the  most  pregnant  witli 
important  conseqaonccs.  When  we  contemplate  the  constituents  of  the 
planetary  system  from  the  point  of  view  which  this  relation  afford."  us,  it 
is  no  longer  mere  analogy  which  strikes  us  —  no  longer  a  general  resem- 
blance among  them,  as  individuals  independent  of  each  other,  and  circula- 
ting about  the  sun,  each  according  to  its  own  peculiar  nature,  and  con- 
nected with  it  by  its  own  peculiar  tie.  The  resemblance  is  now  perceived 
to  be  a  true  family  likeness ;  they  are  bound  up  in  one  chain  —  inter- 
woven in  one  web  of  mutual  relation  and  harmonious  agreement  —  sub- 
jected to  one  pervading  influence,  which  extends  from  the  centre  to  the 
farthest  limits  of  that  great  system,  of  which  all  of  them,  the  earth 
included,  must  henceforth  be  regarded  as  members. 

(489.)  The  laws  of  elliptic  motion  about  the  sun  as  a  focus,  and  of  the 
equable  description  of  areas  by  lines  joining  the  sun  and  planets,  were 
originally  established  by  Kepler,  from  a  consideration  of  the  observed 
motions  of  Mars ;  and  were  by  him  extended,  analogically,  to  all  the  other 
planets.  However  precarious  such  an  extension  might  then  have  ap- 
peared, modern  astronomy  has  completely  verified  it  as  a  matter  of  fact, 
by  the  general  coincidence  of  its  results  with  entire  series  of  observations 
of  the  apparent  places  of  the  planets.  These  are  found  to  accord  satis- 
factorily with  the  assumption  of  a  particular  ellipse  for  each  planet,  whose 
magnitude,  degree  of  exccntricity,  and  situation  in  space,  are  numerically 
assigned  in  the  synoptic  table  before  referred  to.  It  is  true,  that  when 
observations  are  carried  to  a  high  degree  of  precision,  and  when  each 
planet  is  traced  through  many  successive  revolutions,  and  its  history  car- 

'  The  expression  of  this  law  of  Kepler  requires  a  slight  modification  when  we  come 
to  the  extreme  nicety  of  numerical  calculation,  for  the  greater  planets,  due  to  tho 
influence  of  their  masses.    This  correction  is  imperceptible  for  the  Earth  and  Man. 


:;Sl 


X .  y' 


260 


OUTLINES  OF  ASTRONOMT. 


t 


m 


i    ¥ 
I    I 


ricd  back,  by  the  aid  of  oalciilations  founded  on  these  data,  for  n:any  cen- 
turies, we  learn  to  regard  the  laws  of  Kepler  as  only  Jirst  approximations 
to  the  much  more  complicated  ones  which  actually  prevail ;  and  that  to 
bring  remote  observations  into  rigorous  and  mathematical  accordance 
with  each  other,  and  at  the  same  time  to  retain  the  extremely  convenient 
nomenclature  and  relations  of  the  ulliptio  system,  it  becomes  necessary 
to  modify,  to  a  certain  extent,  our  verbal  expression  of  the  laws,  and  to 
regard  the  numerical  data  or  elliptic  elements  of  the  planetary  orbits  as 
not  absolutely  permanent,  but  subject  to  a  series  of  extremely  slow  and 
almost  imperceptible  changes.  These  changes  may  bo  neglected  when  wc 
consider  only  a  few  revolutions ;  but  going  on  from  century  to  century, 
and  continually  accumulating,  they  at  length  produce  material  departures 
in  the  orbits  from  their  original  state.  Their  explanation  will  form  the 
subject  of  a  subsequent  (;bapter  j  but  for  the  present  we  must  lay  them 
out  of  consideration,  as  of  an  order  too  minute  to  affect .  the  general  con- 
clusions with  which  we  are  now  concerned.  By  what  means  astronomers 
are  enabled  to  compare  the  results  of  the  elliptic  theory  with  obscrvatioL, 
and  thus  satisfy  themsehes  of  its  accordance  with  nature,  will  bo  ex- 
plained presently. 

(490.)  It  will  first,  however,  be  proper  to  point  out  what  particular 
theoretical  conclusion  is  involved  in  each  of  the  three  laws  of  Kepler, 
coDfeidered  as  satisfactorily  established, — what  indication  each  of  them, 
separately,  affords  of  the  mechanical  forces  prevalent  in  our  system,  aud 
the  mode  in  which  its  parts  are  connected, — and  how,  when  thus  cou- 
hidered,  they  constitute  the  basis  on  w.'ich  the  Newtonian  explanation  of 
the  mechanism  of  the  heavens  is  mainly  supported.  To  begin  with  the 
first  law,  that  of  the  equable  description  of  areas. — Since  ♦:Uo  planets  move 
in  curvilinear  paths,  they  must  (if  they  be  bodies  obeying  the  laws  of 
dynamics)  be  deflected  from  their  otherwise  natural  rectilinear  progress 
hj/  force.  And  from  this  law,  taken  as  a  matter  of  observed  fact,  it  fol- 
lows, that  the  direction  of  such  force,  at  every  point  of  the  orbit  of  each 
planet,  always  passes  through  the  sun.  No  matter  from  what  ultimate 
cause  the  power  which  is  called  gravitatid  originates, — be  it  a  virtue 
lodged  in  the  sun  as  its  receptacle,  or  be  it  pressure  from  without,  or  the 
resultant  of  many  pressures  or  solicitations  of  unknown  fluids,  magnetic 
or  electric  ethers,  or  impulses, — still,  when  finally  brought  under  our  con- 
templation, and  summed  up  into  a  single  resultant  energy — its  direction 
18,  from  every  point  on  all  sides,  towards  the  suns  centre.  As  an  abstract 
dynamical  proposition,  the  reader  will  find  it  demonstrated  by  Newton,  in 
the  first  proposition  of  the  Principia,  with  an  elementary  simplicity  to 
which  we  really  could  add  nothing  but  obscurity  by  amplification,  that 


u 


INTERPRETATION   OF   KEPLER'S   LAWS. 


261 


any  body,  urged  towards  a  certain  central  point  by  a  force  continually 
directed  thereto,  and  thereby  deflected  into  a  curvilinear  path,  will  describe 
about  that  centre  equal  areas  in  equal  times ;  and  vice  versd,  that  such 
equable  description  of  areas  is  itself  the  essential  criterion  of  a  continual 
direction  of  the  acting  force  towards  the  centre  to  which  this  character 
belongs.  The  first  law  of  Kepler,  then  ^ives  us  no  information  as  to  the 
nature  or  intensity  of  the  force  urging  the  planets  to  the  sun ;  the  onlj 
conclusion  it  involves  is,  that  it  does  so  urge  them.  It  is  a  property  of 
orbitual  rotation  under  the  influence  of  central  forces  generally,  and,  as 
such,  we  daily  see  it  exemplified  in  a  thousand  familiar  instances.  A 
simple  experimental  illustration  of  it  is  to  tie  a  bullet  to  a  thin  string, 
and,  having  whirled  it  round  with  a  moderate  velocity  in  a  vertical  plane, 
to  draw  the  CL.d  of  the  string  through  a  small  ring,  or  allow  it  to  coil 
itself  round  the  finger,  or  round  a  cylindrical  rod  held  very  firmly  in  a 
horizontal  position.  The  bullet  will  then  approach  the  centre  of  motion 
in  a  spiral  line ;  and  the  increase  not  only  of  its  angular  but  of  its  linear 
velocity,  and  the  rapid  diminution  of  its  periodic  time  when  near  the 
centre,  will  express,  more  clearly  than  any  words,  the  compensation  by 
which  its  uniform  description  of  areas  is  maintained  under  a  constantly 
diminishing  distance.  If  the  motion  be  reversed,  and  the  thread  allowed 
to  uncoil,  beginning  with  a  rapid  impulse,  the  velocity  will  diminish  by 
the  same  degrees  as  it  before  increased.  The  increasing  rapidity  of  a 
dancer's  py'roMeWe,  as  he  draws  in  his  limbs  and  straightens  his  whole  per- 
son, so  as  to  bring  every  part  of  his  frame  as  near  as  possible  to  the  axis 
of  his  motion,  is  another  instance  where  the  connection  of  the  observed 
effect  with  the  central  force  exerted,  though  equally  real,  is  much  less 
obvious. 

(491.)  The  second  law  of  Kepler,  or  that  which  asserts  that  the  planets 
describe  ellipses  about  the  sun  as  their  focus,  involves,  as  a  consequence, 
the  law  of  solar  gravitation  (so  be  it  allowed  to  call  the  force,  whatever  it 
be,  which  urges  them  towards  the  sun)  as  exerted  on  each  individual 
planet,  apart  from  all  connection  with  the  rest.  A  straight  line,  dynamic- 
ally speaking,  is  the  only  path  which  can  be  pursued  by  a  body  absolutely 
free,  and  under  the  action  of  no  external  force.  All  deJkcUon  into  a 
curve  is  evidence  of  the  exertion  of  a  force  ;  and  the  greater  the  deflection 
in  equal  times,  the  more  intense  the  force.  Deflection  from  a  straight 
line  is  only  another  word  for  curvature  of  path ;  and  as  a  circle  is  char- 
acterized by  the  uniformity  of  its  curvatures  in  all  its  parts — so  is  every 
other  curve  (as  an  ellipse)  characterized  by  the  particular  law  which  regu- 
lates the  increase  and  diminution  of  its  curvature  as  we  advance  along  its 
circumference.     The  deflecting  force,  then,  which  continually  benda  a 


(  hh 


i'  fy. 


m 


262 


OUTLINES   OP  ASTRONOMY. 


iqoving  body  into  a  curve,  Juay  be  ascertained,  provided  jts  direction,  in 
the  first  place,  and,  secondly,  the  law  of  curvtture  of  tbo  curve  itself,  be 
known.  Both  these  enter  as  elements  into  tbo  oxpresaiou  of  the  force. 
A  body  may  describe,  for  instance,  an  ellipse,  under  u  great  variety  of 
dispositions  of  tbo  acting  forces  :  it  may  glide  along  it,  for  example,  as  a 
bead  upon  a  polieht  I  wire,  bent  into  an  elliptic  form ;  in  which  case  the 
acting  force  is  always  perpendicular  to  the  wire,  and  the  velocity  is  uni- 
form. In  this  case  the  /oire  is  directed  to  no  fixed  centre,  and  there  is 
no  equable  description  of  areas  at  all.  Or  it  may  describe  it  as  wc  may 
see  done,  if  we  suspend  a  ball  by  a  vcri/  long  string,  and,  drawing  it;  u 
little  aside  from  the  perpendicular,  throw  it  round  with  a  gentle  impulse. 
In  thi.s  case  the  acting  force  is  directed  to  the  centre  of  the  ellipse,  about 
which  areas  are  de^scribed  equally,  and  io  which  a  force  proporlional  to 
the  distance  (the  decomposed  result  of  terrestrial  gravity)  perpetually 
urges  if..*  This  is  at  once  a  very  easy  experiment,  and  a  very  instructive 
one,  and  we  shall  again  refer  to  it.  In  the  case  before  us,  of  an  ellipse 
described  by  the  action  of  a  force  directed  to  the  focm,  the  steps  of  the 
investigation  of  the  law  of  force  arc  these :  1st,  The  law  of  the  areas  de- 
termines the  actual  velocitt/  of  the  revolving  body  at  every  point,  or  tbo 
space  really  run  over  by  it  in  a  given  minute  portion  of  time ;  2dly,  The 
law  of  curvature  of  the  ellipse  determines  the  linear  amount  of  deflection 
from  the  tangent  in  thi  dindion  of  the  fuais,  which  corresponds  to  that 
space  so  run  0\'nr ;  3dly,  and  lastly.  The  laws  of  accelerated  motion  de- 
clare that  the  intensity  of  the  acting  force  causing  such  deflection  in,  its 
oion  direction,  is  measured  by  or  proportional  to  the  amount  of  that  de- 
flection, and  may  therefore  be  calculated  in  any  particular  position,  or 
generally  expressed  by  geometrical  or  algebraic  symbols,  as  a  law  inde- 
pendent of  particular  positions,  when  that  deflection  is  so  calculated  or 
expressed.  We  have  here  the  spirit  of  the  process  by  which  Newton  has 
resolved  this  interesting  problem.  For  its  geometrical  detail,  we  must 
refer  to  the  3d  section  of  his  Principia.  We  know  of  no  artificial  mode 
of  imitating  this  species,  of  elliptic  motion ;  though  a  rude  approximation 
to  it — enough,  however,  to  give  a  conception  of  the  alternate  appruach 
and  recess  of  the  revolving  body  to  and  from  the  focus,  and  the  variation 
of  its  velocity — may  be  had  by  suspending  a  small  steel  bead  to  a  tine  and 
very  long  silk  fibre,  and  setting  it  to  revolve  in  a  small  orbit  round  the 
pole  of  a  powerful  cylindrical  magnet,  held  upright,  and  vertically  under 
the  point  of  suspension. 


*  If  th6  suspended  body  be  a  vessel  full  of  fine  sand,  having  a  small  hole  at  \is  bottom 
the  elliptic  trace  of  its  orbit  will  be  left  in  a  sand  streak  on  a  <able  placed  below  it. 
This  neat  illustration  is  due,  to  the  best  of  my  knowledge,  to  Mr.  Babbage. 


INTERPRETATION    OF   KKPLER  i5   LAWS. 


203 


(492.)  Tho  third  law  of  Keplor,  which  connects  the  distances  and 
periods  of  the  planets  by  a  general  rule,  bears  with  it,  as  its  theoretical 
interpretation,  this  important  consequence,  viz.  that  it  is  ono  and  the  same 
force,  modified  only  by  distance  from  tho  sun,  which  retains  all  the  planetn 
in  their  orbits  about  it.  That  the  attraction  of  tho  sun  (if  such  it  be) 
is  exerted  upon  all  tho  bodies  of  our  system  indiflFerently,  without  regard 
to  tho  peculiar  materials  of  which  they  may  consist,  in  the  exact  pro- 
portion of  their  inertieo,  or  quantities  of  matter;  that  it  is  not,  therefore, 
of  tho  nature  of  the  elective  attractions  of  chemistry  or  of  magnetic 
action,  which  is  powerless  on  other  substances  than  iron  and  some  one  or 
two  more,  but  is  of  a  more  universal  character,  and  extends  equally  to  all 
the  material  constituents  of  our  system,  and  (as  we  shall  hereafter  see 
abundant  reason  to  admit)  to  those  of  ti'her  system^  -han  our  own.  This 
law,  important  and  general  as  it  is,  results,  as  the  simplest  of  corollaries, 
from  the  relations  established  by  Newton  in  tho  section  of  the  Priiicqna 
referred  to  (Prop,  xv.),  from  which  proposition  it  results,  that  if  the  earth 
were  taken  from  its  actual  orbit,  and  launched  anew  in  space  at  the  place, 
in  tho  direction,  and  with  the  velocity  of  any  of  the  other  planets,  it 
would  describe  the  very  same  orbit,  and  in  the  same  period,  which  that 
plauot  actually  does,  a  minute  correction  of  the  period  only  excepted, 
arising  from  the  difference  between  the  mass  of  the  earth  and  that  of  the 
planet.  Small  as  the  planets  are  compared  to  the  sun,  some  of  them  are 
not,  as  the  earth  is,  mere  atoms  in  the  comparison.  The  strict  wording 
of  Kepler's  law,  as  Newton  has  proved  in  his  fifty-ninth  proposition,  is 
applicable  only  to  the  case  of  planets  whose  proportion  to  the  central 
body  is  absolutely  inappreciable.  When  this  is  not  the  case,  the  periodic 
time  is  shortened  in  the  proportion  of  the  square  root  of  the  number  ex- 
pressing the  sun's  mass  or  inertiae,  to  that  of  the  sum  of  the  numbers 
expressing  the  masses  of  the  sun  and  planet ;  and  in  general,  whatever 
be  the  masses  of  two  bodies  revolving  round  each  other  under  the  influ- 
ence of  the  Newtonian  law  of  gravity,  the  square  of  their  periodic  time 
will  be  expressed  by  a  fraction  whose  numerator  is  the  cube  of  their  mean 
distance,  i.  e.  the  greater  semi-axis  of  their  elliptic  orb»ij,  aud  whose  de- 
nominator is  the  sum  of  their  masses.  When  one  of  tha  masses  is  in- 
comparably greater  than  the  other,  this  resolves  into  Kepler's  law ;  but 
when  this  is  not  the  case,  the  proposition  thus  generalized  stands  in  lieu 
of  that  law.  In  the  system  of  the  sun  and  planets,  however,  the  numerical 
correction  thus  introduced  into  the  results  of  Kepler's  law  is  too  small  to 
be  of  any  importance,  the  mass  of  the  largest  of  the  planets  (Jupiter) 
being  much  less  than  a  thousandth  purt  of  that  of  the  sun.     We  shall 


i.'i 


!;  ! 


264 


OUTLINES  OF   ASTRONOMY. 


I 


presently,  however,  porceivo  all  the  importanoe  of  thij  ^oueralizutioc, 
when  we  coiue  to  speak  of  the  satellites. 

(493.)  It  will  first,  however,  bo  proper  to  oxplaio  by  what  process  of 
calculation  the  expression  of  a  planet's  clliptio  orbit  by  its  elements  cud  be 
compared  with  observation,  and  how  we  can  satisfy  ourselves  that  the 
numerical  data  contained  in  a  table  of  such  elements  fur  the  whulc  system 
docs  really  exhibit  a  true  picture  of  it,  and  afford  the  means  of  deter- 
mining its  state  at  every  instant  of  time,  by  the  mere  application  of  Kep- 
ler's laws.  Now,  for  each  planet,  it  is  necessary  for  this  purpot;M  to  know, 
lut,  the  magnitude  and  form  of  its  ellipse;  2dly,  the  situation  of  this 
ellipse  in  space,  with  respect  to  the  ecliptic,  and  to  a  fixed  line  drawn 
therein ;  Sdly,  the  local  situation  of  the  planet  in  its  ellipse  at  some  known 
epoch,  and  its  periodic  time  or  mean  angular  velocity,  or,  as  it  is  cuUcJ, 
its  mean  motion. 

(494.)  The  magnitude  and  form  of  an  ellipse  are  determined  by  its 
greatest  length  and  least  breadth,  or  its  two  principal  axes;  but  for  astro- 
nomical uses  it  is  preferable  to  use  the  semi-axis  major  (or  half  the  groatent 
length),  and  the  excentricity  or  distance  of  the  focus  from  the  centre, 
which  last  is  usuolly  estimated  in  parts  of  the  former.  Thus,  an  ellipse, 
whose  length  is  10  and  breadth  8  parts  of  any  scale,  has  for  its  major 
semi-axis  5,  and  for  its  excentricity  3  such  parts ;  but  when  estimated  in 
parts  of  the  semi-axis,  regarded  as  a  unit,  the  excentricity  is  expressed  by 
the  fraction  |. 

(495.)  The  ecliptic  is  the  plane  to  which  an  inhabitant  of  the  earth 
most  naturally  refers  the  rest  of  the  solar  system,  as  a  sort  of  ground- 
plane  ;  and  the  axis  of  its  orbit  might  be  taken  for  a  line  of  departure  in 
that  plane  or  origin  of  angular  reckoning.  Were  the  axis  Jixetf,  this 
would  be  the  best  possible  origin  of  longitudes ;  but  as  it  has  a  motion 
(though  an  excessively  slow  one),  there  is,  in  fact,  no  advantage  in  reck- 
oning from  the  axis  more  than  from  the  line  of  the  equinoxes,  and  astro- 
nomers therefore  prefer  the  latter,  taking  account  of  its  variation  by 
the  effect  of  precession,  and  restoring  it,  by  calculation  at  every  in- 
stant, to  a  fixed',  position.  Now,  to  determine  the  situation  of  the  ellipse 
described  by  a  planet  with  respect  to  this  plane,  three  elements  require 
to  be  known:  —  l,^t,  the  inclination  of  the  plane  of  the  planet's  orbit 
to  the  plane  of  the  ecliptic;  2dly,  the  line  in  which  these  two  planes 
intersect  each  other,  which  of  necessity  passes  through  the  sun,  and 
whose  position  with  respect  to  the  line  of  the  equinoxes  is  therefore 
given  by  stating  its  longitude.  This  line  is  called  the  line  of  the 
nodes.  When  the  planet  is  in  this  Hue,  in  the  act  of  passing  from  the 
south  to  the  north  side  of  the  ecliptic,  it  is  in  its  ascending  node,  and 


f  ! 


ELEMENTS   OF   A   PLANET  3   OllDIT. 


265 


it4  longitude  at  that  moment  is  tho  element  called  the  longltwh  nf  tlui 
node.  TLcso  two  data  determine  the  situation  of  the  2>lane  of  tho  orbit; 
and  there  only  remains,  for  tho  complete  determination  of  tho  situation 
of  the  planet's  ellipse,  to  know  how  it  is  placed  in  that  piano,  which 
(since  its  focus  is  necessarily  in  the  sun)  is  ascertained  by  stating  tho 
hngilade  of  its  perihelion,  or  the  place  which  tho  extremity  of  tho  axis 
nearest  the  sun  occupies,  when  orthographically  projected  on  tho  ecliptic. 

(49B.)  Tho  dimensions  and  situation  of  the  planet's  orbit  thus  dotor- 
luincd,  it  only  remains,  for  a  complete  acquaintance  with  its  history,  to 
(leterinine  the  circumstances  of  its  motion  in  the  orbit  so  precisely  fixed. 
Now,  for  this  purpose,  all  that  is  needed  is  to  know  the  moment  of  time 
vrbeu  it  is  either  at  the  perihelion,  or  at  any  other  precisely  determined 
point  of  its  orbit,  and  its  whole  period ;  for  these  being  known,  the  law 
of  the  areas  determines  the  place  at  every  other  instant.  This  moment  is 
called  (when  the  perihelion  is  the  point  chosen)  the  perihelion  passage, 
or,  when  some  point  of  the  orbit  is  fixed  upon,  without  special  reference 
to  the  perihelion,  the  epoch. 

(497.)  Thus,  then,  we  have  seven  particulars  or  elements,  which  must 
be  numerically  stated,  before  we  can  reduce  to  calculation  the  state  of  the 
system  at  any  given  moment.  But,  these  known,  it  is  easy  to  ascertain 
the  apparent  positions  of  each  planet,  as  it  would  be  seen  from  tho  sun,  or 
is  seen  from  the  earth  at  any  time.  The  former  is  called  the  heliocentriCf 
the  latter  the  geocentric,  place  of  the  planet. 

(498.)  To  commence  with  the  heliocentric  places.  Let  S  represent 
the  sun  J  PAN  the  orbit  of  the  planet,  being  an  ellipse,  having  the  S  in 
its  focus,  and  A  for  its  perihelion ;  and  letp  «  N  T  represent  the  projection 
of  the  orbit  on  the  plane  of  tho  ecliptic,  intersecting  the  line  of  equinoxes 

Fig.  71. 


r\ 


■\ 


j:!i 


S  r  in  r,  which,  therefore,  is  the  origin  of  longitudes.  Then  will  S  N  be 
the  line  of  nodes ;  and  if  we  suppose  B  to  lie  on  the  south,  and  A  on  the 
north  side  of  the  ecliptic,  and  the  direction  of  the  planet's  motion  to  be 
from  B  to  A,  N  will  be  the  ascending  node,  and  the  angle  T  S  N  the  lon- 
gitude of  the  node.  In  like  manner,  if  P  be  the  place  of  the  planet  at 
any  time,  and  if  it  and  the  perihelion  A  be  projected  on  the  ecliptic,  upon 
the  points  p,  a,  the  angles  T  Sp,  T  S  a,  will  be  the  respective  heliocentrio 


ill 
■!' 


M 


vi  f 


n 


13^ 


266 


OUTLINES   OF  ASTRONOMY. 


longitudes  of  the  planet  and  of  the  perihelion,  the  former  of  which  is  to 
be  determined,  and  the  latter  is  one  of  the  given  element.  Lastly,  the 
angle  p  S  P  is  the  heliocentric  latitude  of  the  planet,  which  is  also  required 
to  be  known. 

(499.)  Now,  the  time  being  given,  and  also  the  moment  of  the  planet's 
passing  the  perihelion,  the  interval,  or  the  time  of  describing  the  portion 
A  P  of  the  orbit,  is  given,  and  the  periodical  time,  and  the  whole  area  of 
the  ellipse  being  known,  the  law  of  proportionality  of  areas  to  the  times 
of  their  description  gives  the  magnitude  of  the  area  ASP.  From  t'  - 
it  is  a  problem  of  pure  geometry  to  determine  the  corresponding  anyk 
ASP,  which  is  called  the  planet's  true  anomaly.  This  problem  is  of 
the  kind  called  transcendental,  and  has  been  resolved  by  a  great  variety 
of  processes,  some  more,  some  less  intricate.  It  offers,  however,  no 
peculiar  difficulty,  and  is  practically  resolved  with  great  facility  by  the 
help  of  tables  constructed  for  the  purpose,  adapted  to  the  case  of  each 
particular  planet.' 

(500.)  The  true  anomaly  thus  obtained,  the  planet's  angular  dipttmce 
from  the  node,  or  the  angle  N  S  P,  is  to  be  found.  Now,  the  longitudes  of 
the  perihelion  and  node  being  respectively  T  a  and  t  N,  which  are  given, 
their  difference  a  N  is  also  given,  and  the  angle  N  of  the  spherical  right- 
angled  triangle  A  N  a,  being  the  inclination  of  the  plane  of  the  orbit  to 
the  ecliptic,  is  known.  Hence  we  calculate  the  arc  N  A,  or  the  angle 
N  S  A,  whichj  added  to  ASP,  gives  the  angle  N  S  P  required.  And 
from  this,  regarded  as  the  measure  of  the  arc  N  P,  forming  the  hypothe- 
nuse  of  the  right-angled  spherical  triangle  P  N  p,  whose  angle  N,  as 
before,  is  known,  it  is  easy  to  obtain  the  other  two  sides,  N  p  and  Vp. 
The  latter,  being  the  measure  of  the  angle  ^  S  P,  expresses  the  planet's 
heliocentric  latitude ;  the  former  measures  the  angle  N  S  j[>,  or  the  planet's 
distance  in  longitude  from  its  node,  which,  added  to  the  known  angle 
T  S  N,  the  longitude  of  the  node,  gives  the  heliocentric  longitude.    This 

'  It  will  readily  be  understood,  that,  except  in  the  case  of  uniform  circular  motion, 
an  equable  description  of  areat  about  any  centre  is  incompatible  with  un  equnhic  de- 
scription of  anglet.  The  object  of  the  problem  in  the  text  is  to  pass  from  the  area 
supposed  known,  to  the  angle,  supposed  unknown  :  in  other  words,  to  derive  the  true 
amount  of  angular  motion  from  the  perihelion)  or  the  true  anomaly  from  what  is  tech- 
nically called  the  mean  anomcly,  that  is,  the  mean  angular  motion  which  would  have 
been  performed  had  the  motion  in  angle  been  uniform  instead  of  the  motion  i«  area. 
It  happens  fortunately,  that  this  is  the  simplest  of  all  problems  of  the  transcendental 
kind,  and  can  be  resolved,  in  the  most  difficult  case,  by  the  rule  of  "  false  position," 
or  trial  and  error,  in  a  very  few  minutes.  Nay,  it  may  even  be  resolved  instantly  on 
inspection  by  a  simple  and  easily  constructed  piece  of  mechanism,  of  which  the  reader 
may  see  a  description  in  the  Cambridge  Philosophical  Transactions,  vol.  iv.  p.  425,  bv 
the  author  of  this  work. 


-5f"  . 


GEOCENTRIC  PLACE  OF  A  PLANET. 


267 


process,  however  circuitous  it  may  appear,  when  once  woU  understood  may 
be  gone  through  numerically  by  the  aid  of  the  usual  logarithmic  and  tri- 
gonometrical tables,  in  little  more  time  than  it  will  have  taken  the  reader 
to  peruse  its  description. 

(501.)  The  geocentric  differs  from  the  heliocentric  place  of  a  planet  by 
reason  of  that  parallactic  change  of  apparent  situation  which  arises  from 
the  earth's  motion  in  its  orbit.  Were  the  planet's  distances  as  vast  as 
those  of  the  stars,  the  earth's  orbitual  motion  would  be  insensible  when 
viewed  from  them,  and  they  would  always  appear  to  us  to  hold  the  same 
relative  situations  among  the  fixed  stars,  as  if  viewed  from  the  sun,  i.  e. 
they  would  then  be  seen  in  their  heliocentric  places.  The  difference,  then, 
between  the  heliocentric  and  geocentric  places  of  a  planet  is,  in  fact,  the 
game  thing  with  its  parallax,  arising  from  the  earth's  removal  from  the 
centre  of  the  system  and  its  annual  motion.  It  follows  from  this,  that 
the  first  step  towards  a  knowledge  of  its  amount,  and  the  consequent 
determination  of  the  apparent  place  of  each  planet,  as  referred  from  the 
earth  to  the  sphere  of  the  fixed  stars,  must  be  to  ascertain  the  proportion 
of  its  linear  distances  from  the  earth  and  from  the  sun,  as  compared  with 
the  earth's  distance  from  the  sun,  and  the  angular  positions  of  all  three 
with  respect  to  each  other. 

(502.)  Suppose,  therefore,  S  to  represent  the  sun,  E  the  earth,  and  P 
the  planet  j  S  T  the  line  of  equinoxes,  T  E  the  earth's  orbit,  and  P  p 
a  perpendicular  let  fall  from  the  planet  on  the  ecliptic.  Then  will  the 
angle  S  P  E  (according  to  the  general  notion  of  parallax  conveyed  in  art. 
69)  represent  the  parallax  of  the  planet  arising  from  the  change  of  station 

Fig.  72. 


•M^ 


!,(    ! 


from  S  to  E ;  E  P  will  be  the  apparent  direction  of  the  planet  seen  from 
E ;  and  if  S  Q  be  drawn  parallel  to  E  p,  the  angle  T  S  Q  will  be  the  geo- 
centric longitude  of  the  planet,  while  T  S  E  represents  the  heliocentric 
longitude  of  the  earth,  T  Sp  that  of  the  planet.  The  former  of  these, 
T  S  E,  is  given  by  the  solar  table ;  the  latter,  T  S  ^,  is  found  by  the  pro- 
cess above  described  (art.  500).  Moreover,  S  P  is  the  radius  vector  of 
the  planet's  orbit,  and  S  E  that  of  the  earth's,  both  of  which  are  determined 


268 


OUTLINES  OF  ASTRONOMY. 


;■  ' 


:|      -        i 

h 


n   'f 


from  the  known  dimensions  of  their  respective  ellipses,  and  the  places  of 
the  bodies  in  them  at  the  assigned  time.  Lastly,  the  angle  P  8  p  is  the 
planet's  heliocentric  latitude. 

(503.)  Our  object,  then,  is,  from  all  these  data,  to  determine  the  angle 
T  S  Q,  and  P  E  p,  which  is  the  geocentric  latitude.  The  process,  then, 
will  stand  as  follows  : — 1st,  In  the  triangle  S  Fp,  right-angled  &tp,  given 
S  P,  and  the  angle  P  Sp  (the  planet's  radius  vector  and  heliocentric  lati- 
tudeV  find  Sp  and  P^;  2dly,  In  the  triangle  SEp,  given  Sp  (just 
found),  S  E  (the  earth's  radius  vector),  and  the  angle  E  S  p  (the  difference 
of  heliocentric  longitudes  of  the  earth  and  planet),  find  the  angle  S  p  E, 
and  the  side  E  p.  The  former  being  equal  to  the  alternate  angle  p  S  Q, 
is  the  pi  rallactic  removal  of  the  planet  in  longitude,  which,  added  to  r  S/), 
gives  its  nreocentric  longitude.  The  latter,  l^p  (which  is  called  the  curtate 
distance  of  the  planet  from  the  earth),  gives  at  once  the  geocentric  lati- 
tude,  by  means  of  the  right-angled  triangle  P  E^,  of  which  E^  and  J*p 
are  known  sides,  and  the  angle  P  E  ^  is  the  geocentric  latitude  sought, 

(504.)  The  calculations  required  for  these  purposes  are  nothing  hut 
the  most  ordinary  processes  of  plane  trigonometry ;  and,  though  some- 
what tedious,  are  neither  intricate  nor  difiicult.  When  executed,  how- 
ever, they  afford  us  the  means  of  comparing  the  places  of  the  planets 
actually  observed  with  the  elliptic  theory,  with  the  utmost  exactness,  and 
thus  putting  it  to  the  severest  trial ;  and  it  is  upon  the  testimony  of  such 
computations,  so  brought  into  comparison  with  observed  facts,  that  we 
declare  that  theory  to  be  a  true  representation  of  nature. 

(505.)  The  planets  Mercury,  Venus,  Mars,  Jupiter,  and  Saturn,  have 
been  known  from  the  earliest  ages  in  which  astronomy  has  been  culti- 
vated. Uranus  was  discovered  by  Sir  W.  Herschel  in  1781,  March  13th, 
in  the  course  of  a  review  of  the  heavens,  in  which  every  star  visible  in  a 
telescope  of  a  certain  power  was  brought  under  close  examination,  when 
the  new  planet  was  immediately  detected  by  its  disc,  under  a  high  magni- 
fying power.  It  has  since  been  ascertained  to  have  been  observed  on 
many  previous  occasions,  with  telescopes  of  insuflScient  power  to  show  its 
disc,  and  even  entered  in  catalogues  as  a  star ;  and  some  of  the  observa- 
tions which  have  been  so  recorded  have  been  used  to  improve  and  extend 
our  knowledge  of  its  orbit,  The  discovery  of  the  ultra-zodiacal  planets 
dates  from  the  first  day  of  1801,  when  Ceres  was  discovered  by  Piazzi,  at 
Palermo;  a  discovery  speedily  followed  by  those  of  Juno  by  professor 
Harding,  of  Gbttingen,  in  1804  j  and  of  Pallas  and  Vesta,  by  Dr.  Olbers, 
of  IJremen,  in  180*2  and  1807  respectively.  It  is  extremely  reniarkablfi 
that  this  important  addition  to  our  system  had  been  in  some  sort  surmised 
as  a  thing  not  unlikely,  on  the  ground  that  the  interval  between  the  orbit 


ORDER  AND   DISCOVERY  OF  THE   PLANETS. 


269 


of  Mercury  and  the  other  planetary  orbits,  go  on  doubling  as  wc  recede 
from  the  sue,  or  nearly  so.  Thus,  the  interval  between  the  orbits  of  the 
Earth  and  Mercury  is  nearly  twice  that  between  those  of  Vcnua  and  Mer- 
cury; that  between  the  orbits  of  Mars  and  Mercur}'^  nearly  twice  that 
between  the  Earth  and  Mercury :  and  so  on.  The  interval  between  the 
orbits  of  Jupiter  and  Mercury,  however,  is  much  too  great,  and  would 
form  an  exception  to  this  law,  which  is,  however,  again  resumed  in  the 
case  of  the  three  planets  next  in  order  of  remoteness,  Jupiter^  Saturn,  and 
Uranus.  It  was  therefore  thrown  out,  by  the  late  professor  Bode,  of  Ber- 
lin,' as  a  possible  surmise,  that  a  planet  not  then  yet  discovered  might 
exist  between  Mars  and  Jupiter:  and  it  may  easily  be  imagined  what 
was  the  astonishment  of  astronomers  on  finding  not  only  one,  but  four 
planets,  differing  greatly  in  all  the  other  elements  of  their  orbits,  but 
agreeing  very  nearly,  both  inter  se,  and  with  the  above  stated  empirical 
law,  in  respect  of  their  mean  distances  from  the  sun.  No  account,  d  jm- 
ori  or  from  theory,  was  to  be  giveu  of  this  singular  progression,  which  is 
not,  like  Kepler's  laws,  strictly  exact  in  numerical  verification ;  but  the 
circumstances  we  have  just  mentioned  tended  to  create  a  strong  belief  that 
it  was  something  beyond  a  mere  accidental  coincidence,  and  bore  reference 
to  the  essential  structure  of  the  planetary  system.  It  was  even  conjec- 
tured that  the  ultra-zodiacal  planets  are  fragments  of  some  greater  planet 
which  formerly  circulated  in  that  interval,  but  which  has  been  blown  to 
atoms  by  an  explosion ;  an  idea  countenanced  by  the  exceeding  minute- 
ness of  these  bodies  which  present  discs ;  and  it  was  argued  that  in  that 
case  innumerable  more  such  fragments  must  exist  ftvA  might  come  to  be 
hereafter  discovered.  Whatever  may  be  thought  0/  uch  a  speculation  as 
a  physical  hypothesis,  this  conclusion  has  beeu  verilled  to  a  considerable 
extent  as  a  matter  of  fact  by  subsequent  discovery,  the  result  of  a  careful 
and  minute  examination  and  mapping  down  nf  the  smaller  stars  in  and 
near  the  zodiac,  undertaken  with  that  express  ooject.  Zodiacal  cliarts  of 
this  kind,  the  product  of  the  zeal  and  industry  of  many  astronomers,  have 
been  constructed,  in  which  every  star  down  to  the  ninth  or  tenth  magni- 
tude is  inserted,  and  these  stars  being  compared  with  the  actual  stars  of 
the  heavens,  the  intrusion  of  any  stranger  within  their  limits  cannot  f;iil 
to  bo  noticed  when  the  comparison  is  systematically  conducted  The  dis- 
covery of  Astrroa,  and  that  of  Hebe  by  Professor  Hencke,  dp*e  respec- 
tively from  December  8th,  1845,  and  July  1st,  1847;  those  of  Iris  and 


i; 


illtli    i 


f'i 


if 


mm 


,,,,.,.: 


,  I 


I  '  I   r? 


•h 


Hi 


^fff)  • 


'  The  empirical  law  itself,  aa  we  have  above  stated  it,  is  ascribed  by  Voiron,  not  to 
Bode  (who  would  appear,  however,  at  all  events,  to  have  first  drawn  attention  to  this" 
interpretation  of  its  interruption),  but  to  Professor  Titius  of  Wittemberg.  (Voiron. 
Siipplemon*  to  Bailly.) 


270 


OUTLINES   OF  ASTRONOMY. 


1 1-  . 


Flora,  by  Mr.  Hind,  from  August  13tb  and  October  18th,  1847 ;  of  Me- 
tis, by  Mr.  Graham,  from  April  25,  1848 ;  and  of  Hygeia,  by  M.  De 
Gasparis,  April  12th,  1849. 

(506.)  The  discovery  of  Neptune  marks  in  a  signal  manner  the  matu- 
rity of  astronomical  science.  The  proof,  or  at  least  the  urgent  presunip- 
tion  of  the  existence  of  such  a  planet,  as  a  means  of  accounting  (by  its 
attraction)  for  certain  small  irregularities  observed  in  the  motions  of 
Uranus,  was  afforded  almost  simultaneously  by  the  independent  researches 
of  two  geometers,  Messrs.  Adams  of  Cambridge  and  Leverrier  of  Paris, 
who  were  enabled,  from  theory  alone,  to  calculate  whereabouts  it  ought 
to  appear  in  the  heavens,  if  visible,  the  places  thus  independently  calcu- 
lated agreeing  surprisingly.  Within  a  single  degree  of  the  place  assigned 
by  M.  Leverrier's  calculations,  and  by  him  communicated  to  Dr.  Galle  of 
the  Royal  Observatory  at  Berlin,  it  was  actually  found  by  that  astronomer 
on  the  very  first  night  after  the  receipt  of  that  communication,  on  turning 
a  telescope  on  the  spot,  and  comparing  the  starb  in  its  immediate  neigh- 
bourhood with  those  previously  laid  down  in  one  of  the  zodiacal  chirts 
already  alluded  to.'  This  remarkable  verification  of  an  indication  so 
extraordinary  took  place  on  the  23d  of  September,  1846.* 

(507.)  The  mean  distance  of  Neptune  from  the  sun,  however,  so  far 
from  falling  in  with  the  supposed  law  of  planetary  distances  above  men- 
tioned, offers  a  decided  case  of  discordance.  The  interval  between  its 
orbit  and  that  of  Mercury,  instead  of  being  nearly  double  the  interval 
between  those  of  Uranus  and  Mercury,  does  not,  in  fact,  exceed  the  latter 
interval  by  much  more  than  half  its  amount.  This  remarkable  exception 
may  serve  to  make  us  cautious  in  the  too  ready  admission  of  empirical 
laws  of  this  natu'o  to  the  rank  of  fundamental  truths,  though,  as  in  the 
present  instance,  iMy  may  prove  useful  auxiliaries,  and  serve  as  stepping 
stones,  affording  a  temporary  footing  in  the  path  to  great  discoveries.  The 
force  of  this  remark  will  be  more  apparent  when  we  come  to  explain  more 


t 


*  Constructed  by  Dr.  Bremiker,  of  Berlin.     On  reading  the  history  of  this  noble 
discovery   we  are  re«dy  to  exclaim  with  Schill« '  — 

"  Mit  dem  Genius  stpht  die  Natur  in  ewigem  Bunde, 
Was  der  Eine  verspricht  liest^t  die  Andre  gewiss."    • 

'Professor  Challia,  </  the  Cambridge  Observatory,  dire''ting  the  Northumberland 
telescope  of  that  Institution  to  the  place  assignee*  !iy  Mr.  Adanis'e  calculations  and  its 
vicmity,  on  the  4th  and  12th  of  August  1846,  8»w  <\\c  plappt  on  boi/i  those  days,  and 
noted  its  place  (among  those  of  other  stars)  for  r»  -'5*»«ervalio''  He,  however,  posr 
poned  the  compariion  of  the  place*  ol.mrved,  »n0  not  |w«««-*«mg  Dr  Breniiker's 
chart  (which  would  have  at  once  indicated  tb*  fiTMience  '4  an  unmapped  siar,) 
remained  in  ignorance  o^  ite  planet'*  existence  m  a  visible  tt^ittt  till  its  annoutie* 
men!  as  such  by  Dr.  Gctl#. 


PHYSICAL   DESCRIPTION   OF  THE   PLANETS. 


271 


particularly  the  nature  of  the  theoretical  views  which  led  to  the  discovery 
of  Neptune  itself. 

(508.)  We  shall  devote  the  rest  cf  this  chapter  to  an  account  of  the 
physical  peculiarities  and  probable  condition  of  the  several  planets,  so  far 
as  the  former  are  known  by  observation,  or  the  latter  lest  on  probable 
grounds  of  conjecture.  In  this,  three  features  principally  strike  us  as 
necessarily  productive  of  extraordinary  diversity  in  the  provisions  by 
which,  if  they  be,  like  our  earth,  inhabited,  animal  life  must  be  supported. 
These  are,  first,  the  difference  in  tbeir  respective  supplies  of  light  and 
beat  from  the  sun ;  secondly,  the  difference  in  the  intensities  of  the 
gravitating  forces  which  must  subsist  at  their  surfaces,  or  the  different 
ratios  which,  on  their  several  globes,  the  inertiee  of  bodies  must  bear  to 
their  weights ;  and,  thirdly,  the  difference  in  the  nature  of  the  materials 
of  which,  from  what  we  know  of  their  mean  density,  we  have  every 
reason  to  believe  they  consist.  The  intensity  of  solar  radiation  is  nearly 
seven  times  greater  on  Mercury  than  on  the  Earth,  and  on  Uranus  330 
limes  less ;  the  proportion  between  the  two  extremes  being  that  of  upwards 
of  2000  to  1.  Let  any  one  figure  to  himself  the  condition  of  our  globe, 
were  the  sun  to  be  septupled,  to  say  nothing  of  the  greater  ratio  !  or  were 
it  diminished  to  a  seventh,  or  to  a  300th  of  its  actual  power !  Again, 
the  intensity  of  gravity,  or  its  efficacy  in  counteracting  muscular  power 
and  repressing  animal  activity,  on  Jupiter,  is  nearly  two  and  a  half  times 
that  on  the  earth,  on  Mars  not  more  than  one-half,  on  the  Moon  one- 
sixth,  and  on  the  smaller  planets  probably  not  more  than  one-twentieth ; 
giving  a  scale  of  which  the  extremes  are  in  the  proportion  of  sixty  to  one. 
Lastly,  the  density  of  Saturn  hardly  exceeds  one-eighth  of  the  mean 
density  of  the  Earth,  so  that  it  must  consist  of  materials  not  much  heavier 
than  cork.  Now,  under  the  various  combinations  of  elements  so  important 
to  life  as  these,  what  immense  diversity  must  wo  not  admit  in  the  condi- 
tions of  that  great  problem,  the  maintenance  of  animal  and  intellectual 
existence  and  happiness,  which  seems,  so  far  as  we  can  judge  by  what  we 
see  around  us  in  our  own  planet,  and  by  the  way  in  which  every  corner 
of  it  is  crowded  with  living  beings,  to  form  an  unceasing  and  worthy 
object  for  the  exercise  of  the  Benevolence  and  Wisdom  which  preside 
over  all ! 

(509.)  Quitting,  however,  the  region  of  mere  speculation,  we  wi^  now 
show  what  information  the  telescope  affords  us  of  the  actual  condition  of 
the  several  planets  within  its  reach.  Of  Mercury  we  can  see  little  more 
than  that  it  is  round,  and  exhibits  phases.  It  is  too  small,  and  too  much 
'ost  in  the  constant  neighbourhood  of  the  Sun,  to  allow  us  to  make  out 


272 


OUTLINES   OF  ASTRONOMY. 


I 


It « 


more  of  it3  nature.  The  real  diameter  of  Mercury  is  about  8200  milcg : 
its  apparent  diameter  varies  from  5"  to  12".  Nor  does  Venus  offer  any 
remarkable  peculiarities :  although  its  real  diameter  is  7800  miles,  and 
although  it  occasionally  attains  the  considerable  apparent  diameter  of  61" 
which  is  larger  than  that  of  any  other  planet,  it  is  yet  the  most  difficult 
of  them  all  to  define  with  telescopes.  The  intense  lustre  of  its  illumin- 
ated  part  dazzles  the  sight,  and  exaggerates  every  imperfection  of  the  tele- 
scope; yet  we  see  clearly  that  its  surface  is  not  mottled  over  with  permanent 
spots  like  the  Moon ;  we  notice  in  it  neither  mountains  nor  shadows,  but 
a  uniform  brightness,  in  which  sometimes  we  may  indeed  fancy,  or  per- 
haps more  than  fancy,  brighter  or  obscure:  portions,  but  can  seldom  or 
never  rest  fully  satisfied  of  the  fact.  It  is  from  some  observations  of  this 
kind  that  both  Venus  and  Mercury  have  been  concluded  to  revolve  on 
their  axes  in  about  the  same  time  as  the  Earth,  though  in  the  case  of 
Venus,  Bianchini  and  other  more  recent  observers  have  contended  for  a 
period  of  twenty-four  times  that  length.  The  most  natural  conclusion, 
from  the  very  rare  appearance  and  want  of  permatence  in  the  spots,  is, 
that  we  do  not  see,  as  in  tLe  Moon,  the  real  surface  of  these  planets,  but 
only  their  atmospheres,  much  loaded  with  clouds,  and  which  may  serve 
to  mitigate  the  otherwise  intense  glare  of  their  sunshine. 

(510.)  The  case  is  very  different  with  Mars.  In  this  planet  we  fre- 
quently discern,  with  perfect  distinctness,  the  outlines  of  what  may  be 
continents  and  seas.  (See Plate  III.^^.l.,  which  represents  Mars  in  its 
gibbous  state,  as  seen  on  the  16th  of  August,  1830,  in  the  20-feet 
reflector  at  Slough.)  Of  these,  the  former  are  distinguished  by  that 
ruddy  colour  which  characterizea  the  light  of  this  planet  (which  always 
appears  red  and  fiery),  and  indicates,  no  doubt,  an  ochrey  tinge  in  the 
general  soil,  like  what  the  red  sandstone  districts  on  the  Earth  may  pos- 
sibly offer  to  the  inhabitants  of  Mars,  only  more  decided.  Contrasted 
with  this  (by  a  general  law  in  optics),  the  seas,  as  we  may  call  thcra, 
appear  greenish.'  Tbese  spots,  however,  are  not  always  to  be  seen 
equally  distinct,  ;jut,  ^liicn  seen,  they  offer  the  appearance  of  fovras  con- 
siderably definite  and  highly  cL  racteristic,"  brought  successively  into 
view  by  the  rotation  of  the  planet,  from  the  assiduous  obgervatioa  of 


;; 


■  ■ ' 
-i 


'  I  have  noticed  the  phaenomena  described  in  the  text  on  many  occasions,  but  nc  er 
more  distinct  tlian  on  the  occasion  when  the  drawing  wb„  made  from  which  the  figure 
in  Plate  III.  is  engraved. — Author. 

'  Thi!  reader  will  find  many  of  thene  forms  represented  in  Sch«macb>  r's  .istronc- 
mische  NachricMen,  No.  191,  434,  and  in  the  cha:  in  No,  349,  by  Mess  .s.  Beer  and 
Madler. 


JUPITER. 


273 


which  it  has  even  been  found  practicable  to  construct  a  rude  chart  of  the 
surface  of  the  planet.  The  variety  in  the  spots  may  arise  from  the  plantt 
not  being  destitute  of  atmosphere  and  clouds  j  and  what  atlds  greatly  to 
the  probability  of  this  is  the  appearance  of  brilliant  white  spots  at  its 
poles,  —  one  of  which  appears  in  our  figure,  —  which  have  been  conjec- 
tured, with  some  probability,  to  be  snow ;  as  they  disappear  when  they 
have  been  long  exposed  to  the  sun,  and  are  greatest  when  just  emerging 
from  the  long  night  of  their  polar  winter,  the  snow  line  then  extending 
to  about  six  degrees  (reckoned  on  a  meridian  of  the  planet)  from  the  pole. 
By  watching  the  spots  during  a  whole  night,  and  on  successive  nights,  it 
is  found  that  Mars  has  a  rotation  on  an  axis  inclined  about  30"  18'  to  the 
ecliptic,  and  in  a  period  of  24"  37""  23' '  in  the  same  direction  as  the 
Earth's,  or  from  west  to  east.  The  greatest  and  least  apparent  diameters 
of  Mars  ai'e  4"  and  18",  and  its  real  diameter  about  4100  miles. 

(511.)  We  now  come  to  a  much  more  magnificent  planet,  Jupiter,  the 
largest  of  them  uU,  being  in  diameter  no  less  than  87,000  miles,  and  in 
bulk  exceeding  that  of  the  Earth  nearly  1300  times.  It  is,  moreover, 
digniiied  by  the  attendance  of  four  moons,  satellites,  or  secondari/ ph luts, 
as  they  are  called,  which  constantly  accompany  and  revolve  about  it,  jus 
the  Moon  does  round  the  Earth,  and  in  the  Same  direction,  forming  with 
their  principal,  or  primary,  a  beautiful  miniature  system,  entirely  analo- 
gous to  that  greater  one  of  which  their  central  body  is  itself  a  member, 
obeying  the  same  laws,  and  exemplifying,  in  the  most  striking  and  in- 
structive manner,  the  prevalence  of  the  gravitating  power  as  the  rulihg 
principle  of  their  motions ;  of  these,  however,  we  shall  speak  more  at 
large  in  the  next  clin;  ar. 

(51 2.)  The  disc  of  Jupiter  is  aiwny.  observed  to  be  crossed  in  one 
certain  directir^  by  dark  bands  or  belts  presenting  the  appearance,  in 
Plate  III  ';/.  2.;  which  represents  this  planet  as  seen  on  the  23d  of  Sep- 
tember, 1832,  in  the  20-fett  reflector  at  Slough.  These  belts  are,  how- 
ever, Irv  no  means  alike  at  all  times;  they  vary  in  breadth  and  in  situa- 
ticn  ou  the  disc  (though  never  iu  their  general  direetion).  They  have 
even  been  seen  broken  up  and  distributed  over  the  whole  face  of  the 
planet ;  but  this  phaBOomenon  is  extremely  rare.  Branches  running  out 
from  them,  and  sub^y-yistiotis,  as  represented  in  the  figure,  as  well  as  evi- 
dent dark  "ipots,  are  by  no  means  uncommor. ;  and  from  these,  attentively 
watched,  it  is  concluded  that  this  planet  revolves  in  the  surprisingly  short 
period  of  O""  55  ■  50"  (sid.  time),  on  an  axis  perpendicular  to  the  direction 


f  '51  '   .^ 


t4i): 


\  \'\m 


vm 


'■'*y  ft 


>fh  *a 


\   ^\h 


i. 


'  Beer  "^A  Madler,  Astr.  Nachr.  349. 


18 


274 


OUTLINES   OF  ASTRONOMY. 


I   if 


of  tho  belts.  Now,  it  is  very  remarkable,  and  forms  a  most  satisfactory 
comment  on  the  reasoning  by  which  the  spheroidal  figure  of  the  Earth 
has  been  deduced  from  its  diurnal  rotation,  that  the  outline  of  Jupiter's 
disc  is  evidently  not  circular,  but  elliptic,  being  considerably  flattened  in 
the  direction  of  its  axis  of  rotation.  This  appearance  is  no  optical  illu- 
sion,  but  is  authenticated  by  micrometrical  measures,  which  assign  107  to 
100  for  the  proportion  of  the  equatorial  and  polar  diameters.  And  to 
confirm  in  the  strongest  manner,  the  truth  of  those  principles  on  which 
our  former  conclusions  have  been  founded,  and  fully  to  authorize  their 
extension  to  this  remote  system,  it  appears,  on  calculation,  thai  this  is 
really  the  degree  of  oblateness  which  corresponds,  on  those  principles,  to 
tho  dimensions  of  Jupiter,  and  to  the  time  of  his  rotation. 

(513.)  The  parallelism  of  the  belts  to  the  equator  of  Jupiter,  their 
occasional  variations,  and  tho  appearances  of  spots  seen  upon  them,  render 
it  extremely  probable  that  they  subsist  in  the  atmosphere  of  the  planet, 
forming  tracts  of  comparatively  clear  sky,  determined  by  currents  analo- 
gous to  our  trade-winds,  but  of  a  much  more  steady  and  decided  characer, 
as  might  indeed  be  expected  from  the  immense  velocity  of  its  rotation. 
That  it  is  the  comparatively  darker  body  of  the  planet  which  appears  in 
the  belts  is  evident  from  this, — that  they  do  not  come  up  in  all  their 
strength  to  the  edge  of  the  disc,  but  fade  away  gradually  before  they 
reach  it.  (Sec  Plate  III.  fig.  2.)  The  apparent  diameter  of  Jupiter 
varies  from  30"  to  46". 

(514.)  A  still  more  wonderful,  and,  as  it  may  be  termed,  olaborately 
artificial  mechanism,  is  displayed  in  Saturn,  tho  next  in  order  of  remote- 
ness to  Jupiter,  to  which  it  is  not  much  inferior  in  magnitude,  being  about 
79,000  miles  in  diameter,  nearly  1000  times  exceeding  tho  earth  in  bulk, 
and  subtending  an  apparent  angular  diameter  at  the  earth,  of  about  18" 
at  its  mean  distance.  This  stupendous  globe,  besides  being  attended  by 
no  less  than  saven  satellites,  or  moons,  is  surrounded  by  two  broad,  flat, 
extremely  thin  li^gs,  concentric  with  the  planet  and  with  each  other; 
both  lying  in  one  plane,  and  separated  by  a  very  narrow  interval  from 
each  other  throughout  their  whole  circumference,  as  they  arc  from  the 
planet  by  a  much  wider.  Tho  dimensions  of  this  extraordinary  appendage 
are  as  follows ' : — 


'  These  dimensions  'ire  calculated  from  Prof.  Struve's  micrometric  measures,  Mem. 
Art.  Soc.  iii.  301,  with  the  exception  of  the  thickness  of  the  ring,  which  is  concluded 
from  its  total  disappearance  in  1833,  in  a  telescope  which  would  certainly  have  shown, 
B8  a  visible  object,  a  line  of  light  one-twentieth  of  a  second  in  breadth.  The  interval 
of  the  rings  here  stated  is  possibly  somewhat  too  small. 


SATURN. 


275 


Mile*. 

Exterior  diarnetor  of  exterior  ring » 40095  =  176,418 

Interior  ditto 35-28'J=  155,272 

Exterior  dinmoter  of  interior  ring 34'475r  151,690 

Interior  ditto 26  668=  117,339 

Equntorial  diameter  of  the  body 17'99l  =  79,160 

Interval  between  the  plnnot  and  interior  ring 4339=   19,090 

Interval  of  the  rings 0"40S=     1,791 

Thickness  of  the  ringa  not  exceeding =        250 

The  figure  {f(j.  8,  I'lato  III.)  represents  Satura  surrounded  by  its  rings, 
and  having  its  body  striped  with  dark,  belts,  somowhat  sii;  ilar,  but  broader 
and  less  srongly  marked  than  those  of  Jupiter,  and  owing,  doubtless,  to 
a  similar  cause.'  That  the  ring  is  a  solid  opake  substance  is  shown  by  its 
throwing  its  ahadow  na  tho  body  of  the  planet,  on  the  side  nearest  tho 
suu,  and  on  the  other  side  receiving  that  of  the  body,  as  shown  in  the 
figure.  From  tho  parallelism  of  tho  belts  with  the  plane  of  tho  ring  it 
luay  bo  conjectured  that  the  axis  of  rotation  of  the  planet  is  perpendicular 
to  that  plane  j  and  this  conjecture  is  confii-med  by  the  occasional  appearance 
of  extensive  dusky  spots  on  its  sarface,  which  when  watched,  like  the 
spots  on  Mars  or  Jupiter,  indicate  a  rotation  in  10"  29"  17'  about  an  axis 
so  situated. 

(515.)  Tho  axis  of  rotation,  like  that  of  the  earth,  preserves  its  paral- 
lelism to  itself  during  the  motion  of  the  planet  in  its  orbit;  and  the  same 
is  also  the  case  with  tho  ring,  whose  plane  is  constantly  inclined  at  the 
same,  or  very  nearly  the  same,  angle  to  that  of  the  orbit,  and,  therefore, 
to  the  ecliptic,  viz.  28"  11/ j  and  intersects  the  latter  plane  in  a  line, 
which  makes  at  present*  an  angle  with  the  lino  of  equinoxes  of  167°  31'. 
So  that  the  nodes  of  the  ring  lie  in  167'^  31'  and  347°  31'  of  longitude. 
Whenever,  then,  the  planet  happens  to  be  situated  in  one  or  other  of  thesp 
longitudes,  as  at  C,  the  plane  of  the  ring  passes  through  the  sun,  which 
then  illuminates  only  the  edge  of  it.  And  if  tho  earth  at  that  moment 
be  in  F,  it  will  see  the  ring  edgeways,  the  planet  being  in  opposition,  and 
therefore  most  favourably  situated  (cmteris  paribus)  for  observation. 
Under  these  circumstances  the  ring,  if  seen  at  all,  can  only  appear  as  a 
very  narrow  straight  line  of  light  projecting  on  either  side  of  the  body  as 
a  prolongation  of  its  diameter.  In  fact,  it  is  quite  invisible  in  any  but 
telescopes  of  extraordinary  power.**     This  remarkable  phenomenon  takea 

'  The  equatorial  bright  belt  is  generally  well  seen.  The  subdivision  of  the  dark  one 
by  two  narrow  bright  bands  is  seldom  so  distinct  as  represented  in  the  plate. 

*  According  to  Bessel,  the  longitude  of  the  node  of  the  ring  increases  46"*4C2  per 
annum.    In  1800  it  was  166°  53'  8"-9. 

'  Its  disappearance  was  complete  when  observed  with  a  reflector  eighteen  inches  In 
aperture  and  twenty  feet  in  focal  length,  on  the  29th  of  April,  1833,  by  the  author. 


I  ill  pi 


276 


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OUTLINES   OF   ASTRONOMY. 


Fig.  78. 


place  at  intervals  of  fifteen  years  nearly  (being  a  semi-period  of  Saturn  in 
its  orbit).  One  disappearance  at  least  must  t^ike  place  whenever  Satirn 
passes  either  node  of  its  orbit ;  but  three  must  frequently  happen,  and 
two  are  possible.  To  show  this,  suppose  S  to  be  the  sun,  A  B  C  I)  part 
of  Saturn's  orbit  situated  so  as  to  include  the  node  of  the  ring  (at  C)j 
E  F  G  H  the  earth's  orbit :  S  C  the  line  of  the  node;  E B,  G  D  parallel 
to  S  C  toadiiag  the  earth's  orbit  in  E  G;  and  let  the  direction  of  motion 
of  botii  bc'dio-4  be  that  indicated  by  the  arrow.  Then  since  the  ring  pre- 
serve) i;,.j  panllelism,  its  plane  can  nowhere  intersect  the  earth's  orbit,  and 
thetefi/iii  w^  disappearance  can  take  place,  unless  the  plauet  be  between  B 
and  D :  auil,  on  the  other  hand,  a  disappearance  is  possible  (if  the  earth 
be  rightly  situated)  during  the  whole  time  of  the  description  of  the  arc 
B  D.  Now,  since  S  B  or  S  D,  the  distance  of  Saturn  from  the  Suu,  is  to 
S  E  or  S  G,  that  of  the  Earth,  as  9  54  to  1,  the  angle  C  S  D  or  C  S  B  = 
6''  1',  and  the  whole  angle  BSD  =  12"^  2',  which  is  described  by  Saturn 
(on  an  average)  in  359-46  days,  wanting  only  58  days  of  a  complete 
year.  The  Earth  then  d 'Scnbes  very  nearly  an  entire  revolution  within 
the  limits  of  time  when  a  disappearance  is  possible ;  and  since,  in  cither 
half  of  its  orbit  E  F  G  or  G  li  F,  it  may  equally  encounter  the  plane 
of  the  ring,  one  such  encounter  at  least  is  unavoidable  within  the  time 
specified. 

(516.)  Let  G  tt  be  the  arc  of  the  Earth's  orbit  described  from  G  in 
5-8  days.  Then  if,  at  the  moment  of  Saturn's  arrival  at  B,  the  Earth  be 
at  a,  it  will  encounter  the  plane  of  the  ring  advancing  parallel  to  itself 
and  to  B  E  to  meet  it,  somewhere  in  the  quadrant  H  E,  as  at  M,  after 
which  it  will  be  behind  that  plane  (with  reference  to  the  direction  of 
Saturn's  motion)  through  all  the  arc  M  E  F  G  up  to  G;  where  it  will 


DISAPPBARVNCB   OP   SATURN  8   RING. 


277 


again  overtake  it  at  the  very  moment  of  the  planet  quitting  the  arc  B  D. 
In  this  state  of  things  there  will  bo  two  disappearances.  If,  when  Saturn 
is  at  B,  the  Earth  bo  anywhere  in  the  aro  a  J I  E,  it  is  equally  evident 
that  it  will  meet  and  pass  through  the  advancing  plane  of  the  ring  some- 
where in  the  quadrant  H  E,  that  it  will  again  overtake  and  pass  through 
it  somewhere  in  the  semicircle  E  F  G,  and  again  meet  it  in  some  point 
of  the  quadrant  G  II,  so  that  'three  disappearances  will  take  place.  So, 
nlso,  if  the  Earth  bo  at  E  when  Saturn  is  at  B,  the  motion  of  the  Earth 
being  at  that  instant  directly  towards  B,  f'  ■  plane  of  the  ring  will  for  a 
nhort  time  leave  it  behind  j  but  the  gror  lost  being  rapidly  regained 

:is  the  earth's  motion  becomes  obliij  'f  junction,  it  will  soon 

ovortake  and  pass  through  the  plane  -i.   j  part  of  the  quadrant 

E  F,  and  passing  on  through  G  before  ;  ives  at  D,  will  meet  the 

plane  again  in  the  quadrant  G  II.  The  same  will  continue  up  to  a  certain 
point  h,  at  which,  if  the  Earth  b<5  initially  situated,  there  will  be  but  two 
disappearances  —  the  plane  of  the  ring  there  overtaking  the  Earth  for  an 
iustiiiif,  and  being  immediately  again  left  behind  by  it,  to  be  again  en- 
countered by  it  in  G  H.  Finally,  if  the  initial  place  of  the  Earth  (when 
Saturn  is  at  B)  be  in  the  jkc  h  F  a,  there  will  be  but  one  pansage  through 
the  plane  of  the  ring,  viz.,  in  the  semicircle  G  H  E,  the  Earth  being  in 
advance  of  that  plane  throughout  the  whole  of  h  G. 

(51 7-)  The  appearances  will  moreover  be  varied  according  as  the  Earth 
pa8.sc3  from  the  enlightened  to  the  unenlightened  side  of  the  ring,  or  vlve 
versd.  If  C  be  the  ascending  node  of  the  ring,  and  if  the  under  side  of 
the  paper  bd  supposed  south  and  the  upper  north  of  the  ecliptic,  then, 
when  the  Earth  meets  the  plane  of  the  ring  in  the  quadrant  H  E,  it 
passes  from  the  bright  to  the  dark  side  :  where  it  overta/ces  it  in  the 
quadrant  E  F,  the  contrary.  Vice  versd,  when  it  overtakes  it  in  F  G, 
the  transition  is  from  the  bright  to  the  dark  side,  and  the  contrary  where 
it  meets  it  in  G  H.  On  the  other  hand,  when  the  Earth  is  overtaken  by 
the  ring-plane  in  the  interval  E  h,  the  change  is  from  the  bright  to  the 
dark  side.  When  the  dark  side  is  exposed  to  sight,  the  aspect  of  the 
planet  is  very  singular.  It  appears  as  a  bright  round  disc,  with  its  belts, 
&c.,  but  crossed  equatorinlly  by  a  narrow  and  perfect  black  line.  This 
can  never  of  course  happen  when  the  planet  is  more  than  6°  1'  from  the 
node  of  the  ring.  Generally,  the  northern  side  is  enlightened  and  visible 
when  the  heliocentric  longitude  of  Saturn  is  between  173"  32'  and 
341°  30',  and  the  southern  when  between  353"  32'  and  161°  30'.  The 
greatest  opening  of  the  ring  occurs  when  the  planet  is  situated  at  90°  dis- 
tance from  the  node  of  the  ring,  or  in  longitudes  77°  31'  and  257*  31', 


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278 


MZi  OUTLINES  OF  ASTRONOMY.     '^■ 


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and  at  these  points  the  longer  diameter  of  its  apparent  ellipse  is  almost 
exactly  double  the  shorter.  * 

^■-(518.)  It  will  naturallj  be  asked  how  so  stupendous  an  arch,  if  com- 
posed of  solid  and  ponderous  materials,  can  be  sustained  without  collaps- 
ing and  falling  in  upon  the  planet  ?  The  answer  to  this  is  to  be  found  in 
a  swift  rotation  of  the  ring  in  its  own  plane,  which  observation  has  de- 
tected, owing  to  some  portion  of  the  ring  being  a  little  less  bright  than 
others,  and  assigned  its  period  at  10"  32'°  15',  which,  from  what  we  knovr 
of  its  dimensions,  and  of  the  force  of  gravity  in  the  Satumian  system,  is 
very  nearly  the  periodic  time  of  a  satellite  revolving  at  the  same  distance 
as  the  middle  of  its  breadth.  It  is  the  centrifugal  force,  then,  arising 
from  this  rotation,  which  sustains  it;  and,  although  no  observation  nice 
enough  to  exhibit  a  diiTerence  of  periods  between  the  outer  and  inner 
rings  have  hitherto  been  made,  it  is  more  than  probable  that  such  a  diffe- 
rence does  subsist  as  to  place  each  independently  of  the  other  in  a  similar 
state  of  equilibrium. 

(519.)  Although  the  rings  are,  as  we  have  said,  very  nearly  concdntric 
with  the  body  of  Saturn,  yet  micrometrical  measurements  of  extreme 
delicacy  have  demonstrated  that  the  coincident  is  not  mathematically  ex- 
act, but  that  the  centre  of  gravity  of  the  rings  oscillates  round  that  of  the 
body  describing  a  very  minute  orbit,  probably  under  laws  of  much  com- 
plexity. Trifling  as  this  remark  may  appear,  it  is  of  the  utmost  import- 
ance to  the  stability  of  the  system  of  the  rings.  Supposing  them  mathe- 
matically perfect  in  their  circular  form,  and  exactly  concentric  with  the 
planet,  it  is  demonstrable  that  they  would  form  a  system  in  a  state  of  un- 
stable equilibrium,  which  the  slightest  external  power  would  subvert  — 
not  by  causing  a  rupture  in  the  substance  of  the  rings  —  but  by  precipi- 
tating them,  unbroken,  on  the  surface  of  the  planet.  For  the  attraction 
of  such  a  ring  or  rings  on  a  point  or  sphere  excentrically  within  them,  is 
not  the  same  in  all  directions,  but  tends  to  draw  the  point  or  sphere 
towards  the  nearest  part  of  the  ring,  or  away  from  the  centre.  Hence, 
supposing  the  body  to  become,  from  any  cause,  ever  so  little  excentric  to 
the  ring,  the  tendency  of  their  mutual  gravity  is  not  to  correct  but  to 
increase  this  excentricity,  and  to  bring  the  nearest  parts  of  them  together. 
Now,  external  powers,  capable  of  producing  such  excentricity,  exist  in  the 
attractions  of  the  satellites,  as  will  be  shown  in  Chap.  XII. ;  and  in  order 
that  the  system  may  be  stablr,  and  possess  within  itself  a  power  of  resist- 
ing the  first  inroads  of  such  a  tendency,  while  yet  nascent  and  feeble,  aud 
opposing  them  by  an  opposite  or  maintaining  power,  it  has  been  shown 
that  it  is  sufficient  to  admit  the  rings  to  be  loaded  in  some  part  of  their 
circumference,  either  by  some  minute  inequality  of  thickness,  or  by  some 


/; 


EQUILIBRIUM  OF  SATURN'S  RINGS. 


279 


portiona  being  denser  than  others.  Such  a  load  would  gi\)  to  the  whole 
ring  to  which  it  was  attached  somewhat  of  the  character  of  a  heavy  and 
sluggish  satellite  maintainiog  itself  in  an  orbit  with  a  certain  energy  suffi- 
cient to  overcome  minute  causes  of  disturbance,  and  establish  an  average 
bearing  on  its  centre.  But  even  without  supposing  the  existence  of  any 
such  load, — of  which,  after  all,  we  have  no  proof, — and  granting,  in  its 
fall  extent,  the  general  instability  cf  the  equilibrium,  we  think  we  per- 
ceive, in  the  rapid  periodicity  of  all  the  causes  of  disturbance,  a  sufficient 
guarantee  of  its  preservation.  However  homely  be  the  illustration,  we 
can  conceive  nothing  more  apt,  in  every  way,  to  give  a  general  conception 
of  this  maintenance  of  equilibrium  under  a  constant  tendency  to  subver- 
sion, than  the  mode  in  which  a  practised  hand  will  sustain  a  long  pole  in 
a  perpendicular  position  resting  on  the  finger  by  a  continual  and  almoet 
imperceptible  variation  of  the  point  of  support.  Be  that,  however,  as  it 
may,  the  observed  oscillation  of  the  centres  of  the  rings  about  that  of  the 
planet  is  in  itself  the  evidence  of  a  perpetual  contest  between  conserva- 
tive and  destructive  powers  —  both  extremely  feeble,  but  so  antagonizing 
one  another  as  to  preirent  the  latter  from  ever  acquiring  an  uncontrollable 
ascendancy,  and  rushing  to  a  catastrophe. 

(520.)  This  is  also  the  place  to  observe,  that  as  the  smallest  difference 
of  velocity  between  the  body  and  the  rings  must  infallibly  precipitate  the 
latter  on  the  former,  never  more  to  separate,  (for  they  would,  once  in 
contact,  have  attained  a  position  of  stable  equilibrium,  and  be  held  toge- 
ther ever  after  by  an  immense  force ;)  it  follows,  either  that  their  motions 
in  their  common  orbit  round  the  sun  must  have  been  adjusted  to  each 
other  by  an  external  power,  with  the  miuutest  precision,  or  that  the  rings 
niust  have  been  formed  about  the  planet  while  subject  to  their  common 
orbitual  motion,  and  under  the  full  and  free  influence  of  all  the  acting 

(521.)  Several  astronomers  have  suspected,  and  even  conFider  them- 
selves to  have  certainly  observed,  the  rings  of  Saturn  to  be  occasionally, 
at  least,  streaked  with  more  or  less  numerous  dark  lines  parallel  to  the 
decided  black  interval  which  separates  the  two  rings,  which  latter  being 
permanent,  and  seen  equally  and  in  the  same  part  of  the  breadth  on  both 
sides  of  the  ring,  cannot  be  doubted  to  be  a  real  separation.'  ='^    "i  •- 

(522.)  [The  exterior  ring  of  Saturn  is  described  by  many  observers  as 
rather  less  luminous  than  the  interior,  and  the  inner  portion  of  this  latter 

'  The  passage  of  Saturn  ccross  any  considerable  star  would  afford  an  admirable 
opportunity  of  testing  the  reality  of  such  fissures,  as  it  would  flash  in  succession 
through  them.  The  opportunity  of  watching  for  such  occultations  —  when  Saltan 
traverses  the  Milky- Way,  for  instance — should  not  be  neglected. 


>.    ^ 

^ 


280 


JJ5    OUTLINBS  OP  ASTRONOMY.     i>?( 


•  r 


than  its  outer.  On  the  night  of  Nov.  11,  1850,  however,  Mr.  G.  B. 
Bond,  of  the  Harvard  Observatory  (Cambridge,  U.  S.,)  using  the  great 
Fraunhofer  equatorial  of  that  institution,  became  aware  of  a  line  of 
demarcation  between  these  two  portions  so  definite,  and  an  extension 
inwards  of  the  dusky  border  to  such  an  extent  (one  fifth,  hy  measure- 
rnent,  of  the  joint  breadth  of  the  two  old  rings,)  as  to  justify  him  in 
considering  it  as  a  newly-Kliscovered  ring.  On  the  nights  of  the  25th 
and  29th  of  the  same  month,  and  without  knowledge  of  Mr.  Bond's 
observations,  Mr.  Dawes,  at  his  observatory  at  Wateringbury,  by  the  aid 
of  an  exquisite  achromatic  by  Merz,  of  6^  inches  aperture,  observed  the 
very  same  fact,  and  even  more  distinctly,  so  as  to  be  sure  of  a  decidedly 
darker  interval  between  the  old  and  new  rings,  and  even  to  subdivide  the 
latter  into  two  of  unequal  degrees  of  obscurity,  separated  by  a  line  more 
obscure  than  either.J 

,j  (523.)  Of  Uranus  we  see  nothing  but  a  small  round  uniformly  illumi- 
nated disO)  without  rings,  belts,  or  discernible  spots.  Its  apparent  dia- 
meter is  about  4"^  from  which  it  never  variea  much,  owing  to  the  tmall- 
ness  'J  our  orbit  in  comparison  of  its  own.  Its  real  diameter  is  about 
35,000  miles,  and  its  bulk  82  times  that  of  the  earth.  It  is  attended  by 
satellites — ^four  at  least,  probably  five  q^  six — ^whose  orbits  (as  will  be  seen 
in  the  next  chapter)  offer  remarkable  peculiarities. 

(524.)  The  discovery  of  Neptune  is  so  recent,  and  its  situation  in  the 
ecliptic  at  present  so  little  favourable  for  seeing  it  with  perfect  distinctness, 
that  nothing  very  positive  can  be  stated  as  to  its  physical  appearance.  To 
two  observers  it  has  afforded  strong  suspicion  of  being  surrounded  with  a 
ring  very  highly  inclined.  And  from  the  observatio*^ '  of  Mr.  Lassell, 
M.  Otto  Struve,  and  Mr.  Bond,  it  appears  to  be  attend.  'tainly  by  one, 
and  very  probably  by  two  satellites — though  the  existe^oe  of  the  second 
can  hardly  yet  be  considered  as  quite  demonstrated. 

(525.)  If  the  immense  distance  of  Nep'ane  precludes  all  hope  of 
coming  at  much  knowledge  of  its  physical  state,  the  minuteness  of  the 
ultra-zodiacal  phinets  is  no  less  a  bar  into  any  inquiry  into  theirs.  One 
of  them,  Pallas,  has  been  said  to  have  ;',omewhat  of  a  nebulous  or  hazy 
appearance,  indicative  of  an  extensive  and  vaporous  atmosphere,  little 
repressed  and  condensed  by  the  inadequate  gravity  of  so  small  a  mass. 
It  is  probable,  however,  that  the  appearance  in  question  has  originated 
in  some  imperfection  in  the  telescope  employed  or  other  temporary  causes 
of  illusion.  In  Vesta  and  Pallas  only  have  sensible  discs  been  hitherto 
observed,  and  those  only  with  very  high  magnifying  powers.  Vesta  was 
once  seen  by  Schroeter  with  the  naked  eye.  No  doubt  the  most  remark- 
able of  their  peculiarities  must  lie  in  this  opnditioa  oC  their  state.    A 


// 


// 


j:k 


QBNERAL  YIBW  OF  THB  SOLAS   SYSTEM. 


281 


man  placed  on  one  of  them  would  spring  with  ease  60  feet  high,  and 
sastain  no  greater  shock  in  his  descent  than  he  does  on  the  earth  from 
leaping  a  yard.  On  such  planets  ^ants  might  exist;  and  those  enormous 
animals,  which  on  earth  require  the  buoyant  power  of  water  to  counteract 
their  weight,  might  there  be  denizens  of  the  land.  But  of  suuh  speoula- 
tioDS  there  is  no  end. 

(526.)  We  shall  close  this  chapter  with  an  illustration  calculated  to 
convey  to  the  minds  of  our  readers  a  general  impression  pf  the  relative 
magnitudes  and  distances  of  the  parts  of  our  system.  Choose  any  well 
levelled  field  or  bowling-green.  On  it  place  a  globe,  two  feet  in  diameter; 
this  will  represent  the  Sun;  Mercury  will  be  represented  by  a  grain  of 
mastard  seed,  on  the  circumference  of  a  circle  164  feet  in  diameter  for  its 
orbit;  Venus  a  pea,  on  a  circle  284  feet  in  diameter;  the  Earth  also  a 
pea,  on  a  circle  of  430  feet;  Mars  a  rather  large  pin's  head,  on  a  circle 
of  654  feet;  Juno,  Ceres,  Vesta,  and  Pallas,  grains  of  sand,  in  orbits  of 
from  1000  to  1200  feet;  Jupiter  a  moderate-sized  orange,  in  a  circle  nearly 
half  a  mile  across,  Saturn  a  small  orange,  on  a  circle  of  four-fifths  of  a 
mile ;  Uranus  a  full-sized  cherry,  or  small  plum,  upon  the  circumference 
of  a  circle  more  than  a  mile  and  a  half,  and  Neptune  a  good-sized  plum 
on  a  circle  about  two  miles  and  a  half  in  diameter.  As  to  getting  correct 
notions  on  this  subject  by  drawing  circles  on  paper,  or,  still  worse,  from 
those  very  childish  toys  called  orreries,  it  is  out  of  the  question.  To  imi- 
tate the  motions  of  the  planet..,  in  the  above-mentioned  orbits,  Mercury 
must  describe  its  own  diameter  in  41  seconds;  Venus  in  4'"  14*;  the 
Earth,  in  7  minutes;  Mars,  in  4»  48»;  Jupiter,  2"  56";  Saturn,  in  3* 
18*;  Uranus,  in  2'  16";  and  Neptune  in  3'  30".  iks  ?'  ;^i?mn/^u^x.'ii-ffis^ 


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;  Li)' '•'.(«   5>iV    ; 


4t 


■  fi'ik 


>m'' 


■nf-*j 


^/f%    UfT   wa|fi^n(-,T!T 


.1 ' 


ii- 


282 


J«Ktif   OUTLINES  OP  ASTRONOMT.    •'int?rF> 


CHAPTER  X. 


i^m-rf 


.1  ©fetito  1*1*11^:  J 
OP.  THE    SATELLITES.      *^^^Wjfoi:s.r^ 


OP  THE  MOON,  AS  A  SATELLITE  OP  THE  EARTH. — GENERAL  PROXIMITY 
OP  SATELLITES  TO  THEIR  PRIMARIES,  AND  CONSEQUENT  SUBORDINA- 
^  TION  OP  THEIR  MOTIONS.  —  MASSES  OP  THE  PRIMARIES  CONCLUDED 
'''  PROM  THE  PERIODS  OP  THEIR  SATELLITES.  —  MAINTENANCE  OP  KEP- 
'^'  LER's  laws  in  THE  SECONDARY  SYSTEMS. — OP  JUPITER's  SATEL- 
'  LITES. — THEIR  ECLIPSES,  ETC.  —  VELOCITY  OP  LIGHT  DISCOVERED 
BY    THEIR    MEANS.  —  SATELLITES    OP    SATURN — OP    URANUS  —  OP 


NEPTUNE. 


■rfii'/^Mrw 'jifiw-fjfli'  'tiiiMtf  fu  ^vnvishi  |i«>«i*9!lJVt -j»  >^«"!'ii"« 


i 


a  (527.)  In  the  annual  oircnit  of  the  earth  about  the  sun,  it  is  constantly 
attended  by  its  satellite,  the  moon,,  which  revolves  round  it,  or  rather 
both  round  their  common  centre  of  gravity;  nubile  this  centre,  strictly 
speaking,  and  not  either  of  the  two  bodies  thus  connected,  moves  in  an 
elliptic  orbit,  undisturbed  by  their  mutual  action,  just  as  the  centre  of 
gravity  of  a  large  and  small  stone  tied  together  and  flung  into  the  air 
describes  a  parabola  as  if  it  were  a  real  material  substance  under  the 
earth's  attraction,  while  the  stones  circulate  round  it  or  round  each  other, 
as  we  choose  to  conceive  the  matter. 

(528.)  If  we  trace,  therefore,  the  real  curve  actually  described  by 
either  the  moon's  or  the  earth's  centres,  in  virtue  of  this  compound  mo> 
tiou,  it  will  appear  to  be,  not  an  exact  ellipse,  but  an  undulated  curve, 
like  that  represented  in  the  figure  to  article  324,  only  that  the  number 
of  undulations  in  a  whole  revolution  is  but  13,  and  their  actual  deviation 
from  the  general  ellipse,  which  serves  them  as  a  central  line,  is  compara- 
tively very  much  smaller — so  much  so,  indeed,  that  every  part  of  the 
curve  described  by  either  the  earth  or  moon  is  concave  towards  the  sun. 
The  excursions  of  the  earth  on  either  side  of  the  ellipse,  indeed,  are  so 
very  small  as  to  be  hardly  appreciable.  In  fact,  the  oentre  of  gravity  of 
the  earth  and  moon  lies  always  within  the  surface  of  the  earth,  so  that 
the  monthly  orbit  described  by  the  earth's  centre  about  the  commou 
centre  of  gravity  is  comprehended  within  a  space  less  than  the  size  of  the 
earth  itself.    The  effect  is,  nevertheless,  sensible,  in  producing  an  appa- 


OF  THB  MOON  AS  A  SATELLITE. 


28S 


II 


rent  monthly  displaoement  of  the  ain  in  longitude,  of  a  parallactio  kind, 
ffbioh  is  called  the  menstrual  equation ;  whose  greatest  amount  is,  how- 
ever, less  than  the  sun's  horizontal  parallax,  or  than  8-6".  '•^'■>'^'i^. 

(529.)  The  moon,  as  we  have  seen,  is  about  60  radii  of  the  earth  dis- 
tant from  the  centre  of  the  latter.  Its  proximity,  therefore,  to  its  centre 
of  attraction,  thus  estimated,  is  much  greater  than  that  of  the  planets  to 
the  sun ;  of  which  Mercury,  the  neaiest,  is  84,  and  Uranus  2026  solar 
radii  from  its  centre.  It  is  owing  to  this  proximity  that  the  moon 
remains  attached  to  the  earth  as  a  satellite.  Were  it  much  farther,  the 
feebleness  of  its  gravity  towards  the  earth  would  be  inadequate  to  produce 
that  alternate  acceleration  and  retardation  in  its  motion  about  the  sun, 
which  divests  it  of  the  character  of  an  independent  planet,  and  keeps  its 
movements  subordinate  to  those  of  the  earth.  The  one  would  outrun,  or 
be  left  behind  the  other,  in  their  revolutions  round  the  sun  (by  reason  of 
Kepler's  third  law,)  according  to  the  relative  dimensions  of  their  helio- 
centric orbits,  after  which  the  whole  influence  of  the  earth  would  be 
con£aed  to  producing  some  considerable  periodical  disturbance  in  the 
moon's  motion,  as  it  passed  or  was  passed  by  it  in  each  synodical  revolu- 
tion. 

(530.)  At  the  distance  at  which  the  moon  really  is  from  us,  its  gravity 
towards  the  earth  is  actually  less  than  towards  the  sun.  That  this  is  the 
case,  appears  sufficiently  from  what  we  hare  already  stated,  that  the 
moon's  real  path,  even  when  between  the  earth  and  sun,  is  concave 
Uywards  the  latter.  But  it  will  appear  still  more  clearly  if,  from  the 
known  periodic  times '  in  which  the  earth  completes  its  annual  and  the 
moon  its  monthly  orbit,  and  from  the  dimensions  of  those  orbits,  we 
calculate  the  amount  of  deflection,  in  either,  from  their  tangents,  in  equal 
very  minute  portions  of  time,  as  one  second.  These  are  the  versed  sines 
of  ths  arcs  described  in  that  time  in  the  two  orbits,  and  these  are  the 
measures  of  the  acting  forces  which  produce  those  deflections.  If  we 
execute  the  numerical  calculation  in  the  case  before  us,  we  shall  find 
2.233 :  1  for  the  proportion  in  which  the  intensity  of  the  force  which 
retains  the  earth  in  its  orbit  round  the  sun  actually  exceeds  that  by  which 
the  moon  is  retained  in  its  orbit  about  the  earth. 


If 


I 

I 


'  R  and  r  radii  of  two  orbits  (supposed  circular,)  P  and  p  the  periodic  times;  then 
the  arcs  in  question  (A  and  a)  are  to  each  other  as  -=-  to  — ;  and  since  the  versed  sines 
are  as  the  squares  of  the  arcs  directly  and  the  radii  inversely,  these  are  to  each  other 

R         r 

as  —  to  — T  ;  and  in  this  ratio  are  the  forces  acting  on  the  revolving  bodies  i.i  either 

P      F 
case.  .  -  •  • 


284 


OUTLINES  OF  ABXBONOMT.    4^ 


(581.)  Now  tbe  sun  is  809  times  more  remote  from  the  earth  than  the 
moon  is.  And,  as  gravity  increases  as  tbe  squares  of  the  distances  de- 
crease, it  must  follow  that  at  equal  distances,  the  intenuty  of  solar  would 
exceed  that  of  terrestrial  gravity  in  the  above  proportion,  augmented  in 
the  further  ratio  of  the  square  of  400  to  1 ;  that  is,  in  the  proportion  of 
855000  to  1 ;  and  therefore,  if  we  grant  that  the  intensity  of  the  gravi- 
tating energy  is  commensurate  with  tbe  mass  or  inertia  of  tbe  attracting 
body,  we  are  compelled  to  admit  the  mass  of  tbe  earth  to  be  no  more 
than  775^x;7{  of  that  of  tbe  sun. 

(582.)  The  argument  is,  in  fact,  nothing  more  than  a  recapitulation  of 
what  has  been  adduced  in  Chap.  YIII.  (art.  448.)  But  it  is  here  re- 
introduced, in  order  to  show  how  the  mass  of  a  planet  which  is  attended 
by  one  or  more  satellites  can  be  as  it  were  weighed  against  the  sun,  pro- 
vided we  have  learned  from  observation  the  dimensions  of  the  orbits 
described  by  the  planet  about  tbe  sun,  and  by  the  satellites  about  the 
planet,  and  also  the  periods  in  which  these  orbits  are  respectively 
described.  It  is  by  this  method  that  tbe  masses  of  Jupiter,  Sfttom, 
Uranus,  and  Neptune  have  been  ascertained.  (See  Synoptic  Table.) 

(583.)  Jupiter,  as  already  stated,  is  attended  by  four  satellites,  Saturn 
by  seven ;  Uranus,  certainly  by  four,  and  perhaps  by  six ;  and  Neptune 
by  two  or  more.  These,  with  their  respective  jprimaries  (as  the  central 
planets  are  called,)  form  in  each  case  miniature  systems  entirely  analogous, 
in  the  general  laws  by  which  their  motions  are  governed,  to  the  great 
system  in  which  the  sun  acts  the  part  of  tbe  primary,  and  the  planets  of 
its  satellities.  In  each  of  these  systems  the  laws  of  Kepler  are  obeyed, 
in  tbe  sense,  that  is  to  say,  in  which  they  are  obeyed  in  the  planetary 
system  —  approximately,  and  without  prejudice  to  the  effects  of  mutual 
perturbation,  of  extraneous  interference,  if  any,  and  of  that  small  but 
not  imperceptible  correction  which  arises  from  the  elliptic  form  of  the 
central  body.  Their  orbits  are  circles  or  ellipses  of  very  moderate  excen- 
tricity,  tbe  primary  occupying  one  focus.  About  this  they  describe  areas 
very  nearly  proportional  to  tbe  times ;  and  the  squares  of  tbe  periodical 
times  of  all  the  satellites  belonging  to  each  planet  are  in  proportion  to 
each  other  as  tbe  cubes  of  their  distances.  The  tables  at  the  end  of  the 
volume  exhibit  a  synoptic  view  of  the  distances  and  periods  in  these 
several  systems,  so  far  as  they  ore  at  present  known ;  and  to  all  of  them 
it  will  be  observed  that  tbe  same  remark  respecting  their  proximity  to 
their  primaries  holds  good,  as  in  the  case  of  tbe  moon,  with  a  similar 
reoson  for  such  close  connection. 

(534.)  Of  these  systems,  however,  the  only  one  which  has  been  studied 


8ATELLITI8  OF  JUPITER. 


1285 


with  attention  to  all  its  details,  is  that  of  Japiter;  partly  on  account  of 
the  connpiottoas  brillianoy  of  its  four  attendants,  which  are  large  enough 
to  offer  visible  and  measarable  dines  in  telescopes  of  great  power;  but 
more  for  the  sake  of  their  eclipses,  which,  as  they  happen  very  frequently, 
and  are  easily  observed,  afford  signals  of  considerable  use  for  the  determi- 
nation  of  terrestrial  longitudes  (art.  266).  This  method,  indeed,  until 
thrown  into  the  back  ground  by  the  greater  facility  and  exactness  now 
attainable  by  lunar  observations  (art.  267)  was  the  best,  or  rather  the 
only  one,  which  could  be  relied  on  for  great  distances  and  long  intervals. 

(585.)  The  satellites  of  Jupiter  revolve  from  west  to  east  (following  the 
analogy  of  the  planets  and  moon,)  in  planes  very  nearly,  although  not  ex- 
actly, coincident  with  that  of  the  equator  of  the  planet,  or  parallel  to  its 
belts.  This  latter  plane  is  inclined  8°  5'  80"  to  the  orbit  of  the  planet, 
and  is  therefore  but  little  different  from  the  plane  of  the  ecliptic.  Accord- 
ingly, we  see  their  orbits  prelected  very  nearly  into  straight  lines,  in  which 
they  appear  to  oscillate  to  and  fro,  sometimes  passing  before  Jupiter,  and 
casting  shadows  on  his  disc,  (which  are  very  visible  in  good  telescopes, 
like  small  round  ink  spots,  the  circular  form  of  which  is  very  evident,) 
and  sometimes  disappearing  behind  the  body,  or  being  eclipsed  in  its 
shadow  at  a  distance  from  it.  It  is  by  these  eclipses  that  we  are  furnished 
with  accurate  data  for  the  construction  of  tables  of  the  satellites'  motions, 
as  well  as  with  signals  for  determining  differences  of  longitude. 

(536.)  The  eclipses  of  the  satellites,  in  their  general  conception,  are 
perfectly  analogous  to  those  of  the  moon,  but  in  their  detail  they  differ  in 
several  particulars.  Owing  to  the  much  greater  distance  of  Jupiter  from 
the  sun,  and  its  greater  magnitude,  the  cone  of  its  shadow  or  umbra 
(art.  420)  is  greatly  more  elongated,  and  of  far  greater  dimension,  than 
that  of  the  earth.  The  satellites  are,  moreover,  much  less  in  prop  ■■hn 
to  their  primary,  their  orbits  less  inclined  to  its  ecliptic,  and  (compa  sir 
tively  to  the  diameter  of  the  planet)  of  smaller  dimensions,  than  is  the 
case  with  the  moon.  Owing  to  these  causes,  the  three  interior  satellites 
of  Jupiter  pass  through  the  shadow,  and  are  totally  eclipsed,  every  revo- 
lution ;  and  the  fourth,  though,  from  the  greater  inclination  of  its  orbit,  it 
sometimes  escapes  eclipse,  and  may  occasionally  graze  as  it  were  the  border 
of  the  shadow,  and  suffer  partial  eclipse,  yet  does  so  comparatively  seldom,' 
and,  ordinarily  (^leaking,  its  eclipses  happen,  like  those  of  the  rest,  each 

reyolution.  ■   .  ti   --vn-i-r   ^la.     ,  >:  ^i--    .■■  ,.■     ,;;„•   ; i  •..«/.?  I  :v.-.>v>  -^".  :,,^: 

(537.)  These  eclipses,  moreover,  are  not  seen,  as  is  the  C&8&  l^tb:'thdi6 
of  the  moon,  from  the  centre  of  their  motion,  but  from  a  remote  station, 
and  one  whose  situation  with  respect  to  the  line  of  shadow  is  variable. 


286 


OUTLINES  OF  ASTRONOMY. 


>/■ 


*r 


This,  of  ooane,  nukes  no  difference  in  the  time$  of  the  eclipses,  but  s 
very  great  one  in  their  visibility,  and  in  their  apparent  situations  with 
respect  to  the  planet  at  the  moments  of  their  entering  and  quitting  the 
shadow. 

,  (588.)  Suppose  S  to  be  the  sun,  E  the  earth  in  its  orbit  E  F  G  K,  J 
Jupiter,  and  a  b  the  orbit  of  one  of  its  satellites.  The  cone  of  tho  shadow, 
then,  will  have  its  vertex  at  X,  a  point  far  beyond  the  orbits  of  all  the 
satellites ;  and  the  penumbra,  owing  to  the  great  distance  of  the  sun,  and 
the  consequent  smallness  of  the  angle  (about  6'  only)  its  disc  subtends  at 
Jupiter,  will  hardly  extend,  within  the  limits  of  the  satellites'  orbits,  to 
any  perceptible  distance  beyond  the  shadow, — for  which  reason  it  is  not 
represented  in  the  figure.  A  satellite  revolving  from  west  to  east  (in  the 
direction  of  the  arrows)  will  be  eclipsed  when  it  enters  the  shadow  at  a, 
but  not  suddenly,  because,  like  the  moon,  it  has  a  considerable  diameter 
seen  from  the  planet ;  so  that  the  time  elapsing  from  the  first  perceptible 
loss  of  light  to  its  total  extinction  will  be  that  which  it  occupies  in  de- 
scribing aboat  Jupiter  an  angle  equal  to  its  apparent  diameter  as  seen)  from 
the  centre  of  the  planet,  or  rather  somewhat  more,  by  reason  of  the 


1  lifv  'J 

*''^'^''  ''^'-t 


penumbra;  and  the  same  remark  applies  to  its  emergence  at  h.  Now, 
owing  to  the  difference  of  telescopes  and  of  eyes,  it  is  not  possible  to  assign 
the  precUe  moment  of  incipient  obscuration,  or  of  total  extinction  at  a,  nor 
that  of  the  first  glimpse  of  light  falling  on  the  satellite  at  h,  or  the  complete 
recovery  of  its  light.  The  observation  of  an  eclipse,  then,  in  which  only 
the  immersion,  or  only  the  emersion,  is  seen,  is  incomplete,  and  inadequate 
to  afford  any  precise  information,  theoretical  or  practical.  But,  if  both 
the  immersion  and  emersion  can  be  observed  toilh  the  tame  telescope,  and 
hy  the  same  person,  the  interval  of  the  times  will  give  the  duration,  and 
their  mean  the  exact  middle  of  the  eclipse,  when  the  satellite  is  in  the 
line  S  J  X,  i.  e.  the  true  moment  of  its  opposition  to  the  sun.  Such  ob- 
servations, and  such  only,  are  of  use  for  determining  the  periods  and  other 


ECLIPSES  OF  JUPITER'S   SATELLITES. 


287 


purtioalan  of  the  motions  of  the  satollitoB,  and  for  aflfording  data  of  any 
material  use  for  the  calculation  of  terrestrial  longitudes.  The  intervals 
of  the  eclipses,  it  will  be  observed,  give  the  synodic  periods  of  the  satel- 
lites'  revolutions ;  from  vrhioh  their  sidereal  periods  must  be  concluded  by 
the  method  in  art.  418. 

(539.)  It  is  evident,  from  a  mere  inspection  of  our  figure,  that  the 
eclipses  take  place  to  the  west  of  the  planet,  when  the  earth  is  situated 
to  the  west  of  the  line  S  J,  i.  e.  before  the  opposition  of  Jupiter ;  and  to 
the  east,  when  in  the  other  half  of  its  orbit,  or  after  the  opposition. 
When  the  earth  approaches  the  opposition,  the  visual  line  becomes  more 
and  more  nearly  coincident  with  the  direction  of  the  shadow,  and  the  ap- 
parent place  where  the  eclipses  happen  will  be  continually  nearer  and 
nearer  to  the  body  of  the  planet.  When  the  earth  comes  to  F,  a  point 
determined  by  drawing  &  F  to  touch  the  body  of  the  planet,  the  emersions 
will  cease  to  be  visible,  and  will  thenceforth,  up  to  the  time  of  the  oppo- 
sition, happen  behind  the  disc  of  the  planet.  Similarly,  from  the  oppo- 
sition till  the  time  when  the  earth  arrives  at  I,  a  point  determined  by 
drawing  a  I  tangent  to  the  eastern  limb  of  Jupiter,  the  immersions  will 
be  concealed  from  our  view.  When  the  earth  arrives  at  O  (or  H)  the 
immersion  (or  emersion)  will  happen  at  the  very  edge  of  the  visible  disc, 
and  when  between  Gt  and  H  (a  very  small  space),  the  satellites  will  2)a8a 
medipsed  behind  the  limb  of  the  planet. 

(540.)  Both  the  satellites  and  their  shadows  are  frequently  observed  to 
transit  or  pass  across  the  disc  of  the  planet.  When  a  satellite  comes  to 
m,  its  shadow  will  be  thrown  on  Jupiter,  and  will  appear  to  move  across 
it  as  a  black  spot  till  the  satellite  comes  to  n.  But  the  satellite  itself  will 
not  appear  to  enter  on  the  disc  till  it  comes  up  to  the  line  drawn  from  E 
to  the  eastern  edge  of  the  disc,  and  will  not  leave  it  till  it  attains  a  similar 
line  drawn  to  the  western  edge.  It  appears  then  that  the  shadow  will 
precede  the  satellite  in  its  progress  over  the  disc  be/ore  the  opposition  of 
Jupiter,  and  vice  versd.  In  these  transits  of  tho  satellites,  which,  with 
very  powerful  telescopes,  may  be  observed  with  great  precision,  it  fre- 
quently happens  that  the  satellite  itself  is  discernible  on  the  disc  as  a 
bright  spot  if  projected  on  a  dark  belt ;  but  occasionally  also  as  a  dark 
spot  of  smaller  dimensions  than  the  shadow.  This  curiotis  fact  (observed 
by  Schroeter  and  Harding)  has  led  to  a  conclusion  that  certain  of  the 
satellites  have  occasionally  on  their  own  bodies,  or  in  their  atmospheres, 
obscure  spots  of  great  extent.  We  say  of  great  extent ;  for  the  satellites 
of  Jupiter,  small  as  they  appear  to  us,  are  really  bodies  of  considerable 
wze,  as  the  followiuf;  comparative  table  will  show : ' —    ;  .  .^^. 

'  Struve,  Mem.  Art.  Soc.  iii.  301.    '        •     ''    ^  '  '  "^'    "^  "• 


288 


OUTLINES  OF  ASTROKOITT.      .  -X 


Mmd  •pptrant 
dtuiMtor  u  Maa 
from  the  Kkrtb. 

Mmd  appM-cnt 

dUmeUr  u  Mtii 

Arom  Jupiter. 

Dlunetmr  In  milM. 

M*M.I 

Jupiter 

88"-827 
1-017 
0-»ll 
1-488 
1-278 

83'     11" 

17  86 

18  0 
8     48 

87000 
2608 
2088 
8377 
2890 

1-0000000 
0-0000173 
0-0000282 
0-0000886 
0-0000427 

lit  latelltte 

2d            

8d           

4th  

From  which  it  follows,  that  the  firat  satellite  appears  to  a  spectator  on 
>»  Jupiter,  as  large  as  our  moon  to  us;  the  second  and  third  nearly  equal  to 

each  other,  and  of  somewhat  more  than  half  the  apparent  diameter  of  the 
first,  and  the  fourth  about  one  quarter  of  that  diameter.  So  seeo,  the; 
will  frequently,  of  course,  eclipse  one  another,  and  cause  eclipses  of  the 
sun  (the  latter  visible,  however,  only  over  a  very  small  portion  of  the 
planet),  and  their  motions  and  aspects  with  respect  to  each  other  must 
offer  a  perpetual  variety  and  singular  and  pleasing  interest  to  the  iahabi- 
tants  of  their  primary. 

(541.)  Besides  the  eclipses  and  the  transits  of  the  satellites  across  the 
disc,  they  may  also  disappear  to  us  when  not  eclipsed,  by  passing  behind 
the  body  of  the  planet.  Thus,  when  the  earth  is  at  E,  the  immersion  of 
the  satellite-^ill  be  seen  at  a,  and  its  emersion  at  b,  both  to  the  west  of 
the  planet,  after  which  the  satellite,  still  continuing  its  course  in  the  di- 
rection  ab,  will  pass  behind  the  body,  and  again  emerge  on  the  opposite 
side,  after  an  interval  of  occultation  greater  or  less  according  to  the  dis- 
tance of  the  satellite.  This  interval  (on  account  of  the  great  distance  of 
the  earth  compared  with  the  radii  of  the  orbits  of  the  satellites)  varies  but 
little  in  the  case  of  each  satellite,  being  nearly  equal  to  the  time  which 
the  satellite  requires  to  describe  an  arc  of  its  orbit,  equal  to  the  angular 
diameter  of  Jupiter  as  seen  from  its  centre,  which  time,  for  the  several 
satellites,  is  as  follows :  viz.,  for  the  first,  2>'  20" ;  for  the  second,  2''  SG" ;  for 
the  third,  8"  43" ;  and  for  the  fourth,  4"  SG" ;  the  corresponding  diameters 
of  the  planet  as  seen  from  these  respective  satellites  being,  19°  49' ;  12° 
26' J  7°  47';  and  4°  25'.'  Before  the  opposition  of  Jupiter,  these  occul- 
'  tations  of  the  satellites  happen  after  the  eclipses :  after  the  opposition 
(when,  for  instance,  the  earth  is  in  the  situation  K),  the  occul^tions  take 
place  before  the  eclipses.  It  is  to  be  observed  that  owing  to  the  proximity 
of  the  orbits  of  the  first  and  second  satellites  to  the  planet,  both  the  im- 
mersion and  emersion  of  either  of  them  can  never  be  observed  in  any 


'  Laplace,  Mec.  Cel.  liv.  viii,  $  27. 


■■'^^'.■n>-irf^j>  ■'■»/  »>i.:. 


i'si-i  :<'t*)i 


*  These  data  are  taken  approximately  from  Mr.  Woodhouse'a  paper  in  the  supplement 
jn  the  Nautical  Almanack  for  1635. 


it 


n 


ECLIPSES  OF  Jupiter's  satellites. 


289 


■inglo  eclipse,  the  immeraion  boing  coacoaled  by  tho  body,  if  the  planet 
be  post  its  opposition,  the  emersion  if  not  yet  arrived  at  it.  So  also  c. 
the  occultation.  The  oommenoeraent  of  the  ocoultation,  or  tbe  passage 
of  the  satellite  behind  tho  disc,  takes  place  while  obscured  by  the  shadow, 
liofore  opposition,  and  its  ro-emorgonce  after.  All  these  particulars  will 
be  easily  apparent  on  mere  inspection  of  the  figure  (art.  636).  It  is  only 
during  the  short  time  that  the  earth  is  in  the  are  G  H  (i.  0.  between  the 
gaa  and  Jupiter,  that  the  cone  of  the  shadow  converging  (while  that  of 
the  visual  rays  diverges)  behind  the  planet,  permits  their  occultations  to 
be  completely  observed  both  at  ingress  and  egress,  unobsourcd,  tho  eclipses 
being  then  invisible. 

(542.)  An  extremely  singular  relation  subsists  between  the  mean 
angular  velocities  or  mean  motions  (as  they  are  termed)  of  the  three  first 
satellites  of  Jupiter.  If  the  mean  angular  velocity  of  the  first  satellite 
be  added  to  twice  that  of  the  third,  the  sum  will  equal  three  times  that 
of  the  second.  From  this  relation  it  follows,  that  if  from  the  mean  lon« 
gitude  of  the  first  added  to  twice  that  of  the  third,  be  subducted  three 
times  that  of  the  second,  the  remainder  will  always  be  the  same,  or  con- 
stant, and  observation  informs  us  that  this  constant  is  ISO**,  or  two  right 
angles;  so  that,  the  situations  of  any  two  of  them  being  given,  that 
of  the  third  may  be  found.  It  has  been  attempted  to  account  for  this 
remarkable  fact,  on  the  theory  of  gravity  by  their  mutual  action ;  and 
Laplace  has  demonstrated,  that  if  this  relation  were  at  any  one  epoch  ap- 
proximately true,  the  mutual  attractions  of  the  satellites  would,  in  process 
of  time,  render  it  exactly  so.  One  curious  consequence  is,  that  these 
three  satellites  cannot  be  all  eclipsed  at  once ;  for,  in  consequence  of  the 
last-mentioned  relation,  when  the  second  and  third  lie  in  the  same  direc- 
tion from  the  centre,  the  first  must  lie  on  the  opposite;  and  thercforr, 
when  at  such  a  conjuncture  the  first  is  eclipsed,  the  other  two  must  lie 
between  the  sun  and  planet,  throwing  its  shadow  on  the  disc,  and  vice 
versA.      '''f  **--  ."*  !<'^- -tiav.,  ^v.  •■•^^ -^-ykr  "H^iv-J  ;'-?*:,■}*  i'>i^  '>;, '..->•:  i.-  •.£■'■'« 

(543.)  Although,  however,  for  the  above  mentioned  reason,  the  satel- 
lites cannot  be  all  eclipsed  at  once,  yet  it  may  happen,  and  occasionally 
does  so,  that  all  are  either  eclipsed,  occulted,  or  projected  on  the  body,  in 
which  case  they  are,  generally  speaking,  equally  invisible,  since  it  requires 
an  excellent  telescope  to  discern  a  satellite  on  the  body,  except  in  peculiar 
circumstances.  Instances  of  the  actual  observations  of  Jupiter  thus 
denuded  of  its  usual  attendance  and  ofiering  the  appearance  of  a  solitary 
disc,  though  rare,  have  been  more  than  once  recorded.  The  first  occasion 
in  which  this  was  noticed  was  by  Molyneux,  on  November  2d,  (old  style) 
19 


290 


OUTLINES  OF  ASTRONOMY. 


i  I 

i 


1681.'  A  similar  observation  is  recorded  by  Sir  W.  Herschel  as  made 
by  him  on  May  22d,  1802.  The  phaanomenon  has  also  been  observed 
by  Mr.  Wallis,  on  April  15th,  1826 ;  (in  which  case  the  deprivation  con- 
tinued two  whole  hours ;)  and  lastly  by  Mr.  H.  Griesbach,  on  September 
27th,  1843. 

(544.)  The  discovery  of  Jupiter's  satellites,  one  of  the  first  fruits  of 
the  invention  of  the  telescope,  and  of  Galileo's  early  and  happy  idea  of 
directing  its  new-found  powers  to  the  examination  of  the  heavens,  forms 
one  of  the  most  memorable  epochs  in  the  history  of  astronomy.  The 
first  astronomical  solution  of  the  great  problem  of  "  the  lonffitude" — prac- 
tically the  most  important  for  the  interests  of  mankind  which  has  ever 
been  brought  under  the  dominion  of  strict  scientific  principles,  dates 
immediately  from  their  discovery.  The  final  and  conclusive  establish- 
ment of  the  Copemican  system  of  astronomy  may  also  be  considered  as 
referable  to  the  discovery  and  study  of  this  exquisite  miniature  system, 
in  which  the  laws  of  the  planetary  motions,  as  ascertained  by  Kepler, 
and  especially  that  which  connects  their  periods  and  distances,  Tere 
speedily  traced,  and  found  to  be  satisfactorily  maintained.  And  (aa  if  to 
accumulate  historical  interest  on  this  point)  it  is  to  the  observation  of 
their  eclipses  that  we  owe  the  grand  discovery  of  the  aberration  of  light, 
and  the  consequent  determination  of  the  enormous  velocity  of  that  won- 
derful element.     This  we  must  explain  now  at  large. 

(545.)  The  earth's  orbit  being  concentric  with  that  of  Jupiter  and 
interior  to  it  (see  Jig.  art.  586),  their  mutual  distance  is  continually 
varying,  the  variation  extending  from  the  sum  to  the  difference  of  the 
radii  of  the  two  orbits ;  and  the  difference  of  the  greater  and  least  dis- 
tances being  equal  to  a  diameter  of  the  earth's  orbit.  Now,  it  was 
observed  by  Roemer,  (a  Danish  astronomer,  in  1675,)  on  comparing  to- 
gether observations  of  eclipses  of  the  satellites  during  many  successive 
years,  that  the  eclipses  at  and  about  the  opposition  of  Jupiter  (or  its 
nearest  point  to  the  earth)  took  place  too  soon — sooner,  that  is,  than,  by 
calculation  from  an  average,  he  expected  them ;  whereas  those  which  hap- 
pened when  the  earth  was  in  the  part  of  its  orbit  most  remote  from 
Jupiter  were  always  too  late.  Connecting  the  observed  error  in  their 
computed  times  with  the  variation  of  distance,  he  concluded,  that,  to 
make  the  calculation  on  an  average  period  correspond  with  fact,  an  allow- 
ance in  respect  of  time  behoved  to  be  made  proportional  to  the  excess  or 
defect  of  Jupiter's  distance  from  the  earth  above  or  below  its  average 
amount,  and  such  that  a  difference  of  distance  of  one  diameter  of  the 
earth's  orbit  should  correspond  to  16"  26'"6  of  time  allowed.    Specu- 

'Molyneux,  Optics,  p.  271. 


I 


8ATELLITES  OF  SATURN. 


291 


lating  on  the  probable  physical  cause,  he  was  naturally  led  to  think  of  a 
(rradual  instead  of  an  instantaneous  propagation  of  light.  This  explained 
every  particular  of  the  observed  phenomenon,  but  the  velocity  required 
(192000  miles  per  second)  was  so  great  as  to  startle  many,  and,  at  all 
events,  to  require  confirmation.  This  has  been  afforded  since,  and  of  the 
most  unequivoca'  Lludj  by  Bradley's  discovery  of  the  aberration  of  light 
(art.  329.)  The  velocity  of  light  deduced  from  this  last  phaBnomenon 
differs  by  less  than  one-eightieth  of  its  amount  from  that  calculated  from 
the  eclipses,  and  even  this  difference  will  no  doubt  be  destroyed  by  nicer 
and  more  rigorously  reduced  observations.  l 

(546.)  The  orbits  of  Jupiter's  satellites  are  but  little  excentric,  those 
of  the  two  interior,  indeed,  have  no  perceptible  excentricity.  Their 
mutual  action  produces  in  them  perturbations  analogous  to  those  of  the 
planets  alDut  the  sun,  and  which  have  been  diligently  investigated  by 
Laplace  and  others.  By  assiduous  observation  it  has  been  ascertained 
thaf  they  are  subject  to  marked  fluctuations  in  respect  of  brightness,  and 
that  these  fluctuations  happen  periodically,  according  to  their  position 
with  respect  to  the  sun.  From  this  it  has  been  concluded,  apparently 
with  reason,  that  they  turn  on  their  axes,  like  our  moon,  in  periods  equal 
to  their  respective  sidereal  revolutions  about  their  primary. 

(547.)  The  satellites  of  Saturn  have  been  much  less  studied  than  those 
of  Jupiter,  being  far  more  difficult  to  observe.  The  most  distant  has  its 
orbit  materially  inclined  (no  less  than  12°  14')'  to  the  plane  of  the  ring, 
with  which  the  orbits  of  all  the  rest  nearly  coincide.  Nor  is  this  the  only 
circumstance  which  separates  it  by  a  marked  difference  of  character  from 
the  system  of  the  six  interior  ones,  and  renders  it  in  some  sort  an  anoma- 
lous member  of  the  Satumian  system.  Its  distance  from  the  planet's  centre 
exceeds  in  the  proportion  of  nearly  three  to  one  that  of  the  most  distant 
of  all  the  rest,  being  no  less  than  64  times  the  radius  of  the  globe  nf 
Saturn,  a  distance  from  the  primary  to  which  our  own  moon  (at  60  radii) 
offers  the  only  parallel.  Its  variation  of  light  also  in  different  parts  of  its 
orbit  is  very  much  greater  than  the  case  of  any  other  secondary  planet. 
Dominic  Cassini  indeed  (its  first  discoverer,  A.  d.  1671)  found  it  to  disap- 
pear for  nearly  half  its  revolution,  when  to  the  east  of  Saturn,  and  though 
the  more  powerful  telescopes  now  in  use  enable  us  to  follow  it  round  the 
whole  of  its  circuit,  its  diminution  of  light  is  so  great  in  the  eastern  half 
of  its  orbit  as  to  render  it  somewhat  difficult  to  perceive.  From  this  cir- 
cumstance (viz.  from  the  defalcation  of  light  occurring  constantly  on  the 
same  side  of  Saturn  as  seen  from  the  earth,  the  visual  ray  from  which  is 
never  very  oblique  to  the  direction  in  which  the  sun's  light  falls  on  it)  it 

•  Lalande,  Astron.  Art.  3075.  . . .    . . 


292 


OUTLINES   OF  ASTBONOMT. 


is  presumed  with  much  certainty  that  this  satellite  revolves  on  its  axis  in 
the  exact  time  of  rotation  about  the  primary;  as  we  know  to  be  the  case 
with  the  moon,  and  as  there  is  considerable  ground  for  believing  to  be  so 
with  all  secondaries. 

(548.)  The  next  satellite  in  order  proceeding  inwards  (the  first  in  order 
of  discovery ')  is  by  far  the  largest  and  most  conspicuous  of  all,  and  is 
probably  not  much  inferior  to  Mars  in  size.  It  is  the  only  one  of  tho 
number  whose  theory  and  perturbations  have  been  at  all  inquired  into' 
farther  than  to  verify  Kepler's  law  of  the  periodic  times,  which  holds 
good,  mutatis  mutandisy  and  under  the  ret  uisite  reservations,  in  this,  as 
in  the  system  of  Jupiter.  The  three  next  satellites  still  proceediDg 
inwards '  are  very  minute,  and  require  pretty  powerful  telescopes  to  see 
them;  while  the  two  interior  satellites  which  just  skirt  the  edge  of  the 
ring  *  can  only  be  seen  with  telescopes  of  extraordinary  power  and  per- 
fection, and  under  the  most  favourable  atmospheric  circumstances.  At 
the  epoch  of  their  discovery  they  were  seen  to  thread,  like  beads,  the 
almost  infinitely  thin  fibre  of  light  to  which  the  ring  then  seen>  edge- 
ways, was  reduced,  and  for  a  short  time  to  advance  off  it  at  either  end, 
speedily  to  return,  and  hastening  to  their  habitual  concealment  behind 
the  body.* 

(549.)  Owing  to  the  obliquity  of  the  ring  and  of  the  orbits  of  the 
satellites  to   Saturn's  ecliptic,  there  are  no  eclipses,   occultations,  or 

*  By  Huyghens,  March  25,  1655. 

«  By  Bessel,  Astr.  Naehr.  Nos.  193,  214. 

*  Discovered  by  Dominic  Cassini  in  1672  and  1684. 

*  Discovered  by  Sir  William  Herschel  in  1789. 

'  Considerable  confusion  prevails  in  the  nomenclature  of  the  Saturnian  system, 
owing  to  the  order  of  discovery  not  coinciding  with  that  of  distances,  Astronomers 
have  not  yet  agreed  whether  to  call  the  two  interior  satellites  the  6th  and  7th  (reckon- 
ing inward)  and  the  older  ones  the  1st,  2d,  3d,  4th,  and  5th,  reckoning  outward ;  or  to 
commence  with  the  innermost  and  reckon  outwards  from  1  to  7.  This  confusion  has 
been  attempted  to  be  obviated  by  a  mythological  nomenclature,  in  consonance  with 
that  at  length  completely  established  for  the  primary  planets.  Taking  the  names  ot 
the  Titanian  divinities,  the  following  pentameters  afford  an  easy  artificial  memory, 
commencin£  with  the  most  distant. 

;. .      ,   ,        lapetus,  Titan;  Rhea,  Dione,  Tethys;  (pron.  TSthys) 
-  '  EnceladuB,  Mimas — 

It  is  worth  remarking  that  Simon  Marius,  who  disputed  the  priority  of  the  discovery 
of  Jupiter's  satellites  with  Galileo,  proposed  for  them  mythological  names,  viz  :— lo, 
Europa,  Ganymede,  and  Callisto.  The  revival  of  these  names  would  savour  of  a  prefe- 
rence of  Marius's  claim,  w*^<.ch,  even  if  an  absolute  priority  were  conceded  (which  it  is 
not),  wouid  still  leave  GaUleo's  general  claim  to  the  use  of  the  telescope  as  a  means  of 
astronomical  discovery  intact.  But  in  the  case  of  Jupiter's  satellites  there  exists  no 
(tonfusion  to  rectify.  They  are  constantly  referred  to  by  their  numerical  designations 
in  every  almanack. 


//  , 


SATELLITES   OP  URANUS. 


293 


transits  of  these  bodies  of  .  -sir  shadows  across  the  disc  of  their  primary 
(the  interior  ones  excepted),  until  near  the  time  when  the  ring  is  seen 
edgewise,  and  when  they  do  take  place,  their  observation  is  attended  with 
too  much  difficulty  to  be  of  any  practical  use,  like  the  eclipses  of  Jupit«r's 
satellites,  for  the  determination  of  longitudes,  for  which  reason  they 
have  been  hitherto  little  attended  to  by  astronomers. 

(550.)  A  remarkable  relation  subsists  between  the  periodic  times  of  the 
two  interior  satellites  of  Saturn,  and  those  of  the  two  next  in  order  of 
distance;  viz.  that  the  period  of  the  third  (Tethys)  is  double  that  of  the 
first  (Mimas),  and  that  of  the  fourth  (Dione)  double  that  of  the  second 
(Enceladus).  The  coincidence  is  exact  in  either  case  to  about  one-800th 
part  of  the  larger  period. 

(551.)  The  satellites  of  Uranus  require  very  powerful  and  perfect  tele- 
scopes for  their  observation.  Two  arc,  however,  much  more  conspicuous 
than  the  rest,  and  their  periods  and  distances  from  the  planet  have  been 
ascertained  with  tolerable  certainty.  They  are  the  second  and  fourth  of 
those  set  down  in  the  synoptic  table.  Of  the  remaining  four,  whose  ex- 
istence, though  announced  with  considerable  confidence  by  their  original 
discoverer,  could  hardly  be  regarded  as  fully  demonstrated,  two  only  have 
been  hitherto  reobserved ;  viz.  the  first  of  our  table,  interior  to  the  two 
larger  ones,  by  the  independent  observations  of  Mr.  Lassell,'  and  M. 
Otto  Struve,*  and  the  fourth,  intermediate  between  the  larger  ones,  by 
the  former  of  these  astronomers.  The  remaining  two,  if  future  observa- 
tion should  satisfactorily  establish  their  real  existence,  will  probably  be 
found  to  revolvo  in  orbits  exterior  to  all  these. 

(552.)  The  orbits  of  these  satellites  offer  remarkable,  and,  indeed, 
quite  unexpected  and  unexampled  peculiarities.  Contrary  to  the  un- 
broken analogy  of  the  whole  planetary  system  —  whether  of  primaries  or 
secondaries  —  the  planes  of  their  orbits  are  nearly  perpendicular  to  the 
ecliptic,  being  inclined  no  less  than  78°  58'  to  that  plane,  and  ii  these 
orbits  their  motions  are  retrograde ;  that  is  to  say,  their  positioas,  when 
projected  on  the  ecliptic,  instead  of  advancing /rom  west  to  east  round  the 
centre  of  their  primary,  as  is  the  case  with  every  other  planet  and  satel- 
lite, move  in  the  opposite  direction.  Their  orbits  are  nearly  or  quite 
circular,  and  they  do  not  appear  to  have  any  sensible,  or,  at  least,  any 
rapid  motion  of  nodes,  or  to  have  undergone  any  material  change  of  incli- 
nation, in  the  course,  at  least,  of  half  a  revolution  of  their  primary  round 
the  sun.  When  the  earth  is  in  the  plane  of  their  orbits,  or  nearly  so, 
their  apparent  paths  are  straight  lines  or  very  elongated  ellipses,  in  which 

'  September  14th  to  November  9th,  1847. 
»  October  8th  to  December  10th,  1847. 


M 


294 


OUTLINES  OF  iSTRONOMT. 


case  they  become  invisible,  their  feeble  light  being  effaced  by  the  superior 
light  of  the  planet,  long  before  they  come  up  to  its  disc,  so  that  the  ob- 
servation  of  any  eclipses  or  ocoaltations  they  may  undc:<-go  is  quite  out 
of  the  question,  with  our  present  telescopes. 

(553.)  If  the  observation  of  the  satellites  of  Uranus  be  difficult,  those 
of  Neptune,  owing  to  the  immense  distance  of  that  planet,  may  be  readily 
imagined  to  offer  still  greater  difficulties.  Of  the  existence  of  one,  dis- 
covered by  Mr.  Lassell,'  there  can  remain  no  doubt,  it  having  also  been 
observed  by  other  astronomers,  both  in  Europe  and  America.  Accord- 
ing to  M.  Otto  Struve'  its  orbit  is  inclined  to  the  ecliptic  at  the  considera- 
ble angle  of  35° ;  but  whether,  as  in  the  case  of  the  satellites  of  Uranus, 
the  direction  of  its  motion  be  retrograde,  it  is  not  possible  to  say,  until  it 
shall  have  been  longer  observed. 

»0n  July  8th,  1847.  ^-  ■""'''''       -    '■  v 

*  Aatron.  Nachr.  No.  629,  from  his  own  observations,  September  11th  to  Decem- 
ber 20th,  1847. 


.1 


' , !  .       '  il 


•■*!)•    If       -,      -      ,yf  -      '•        ■<-■. 


,'^■1:.   ..T^^'  ■«'.'';•  i  SI   •-■!^  i "    '  • 


OF  COMETS. 


295 


i...  ■•.<'   -y 


CHAPTER  XL 
OF    COMETS. 


QREAT  NUMBER  OF  RECORDED  COMETS.  —  THE  NUMBER  OF  THOSE 
UNRECORDED  PROBABLT  MUCH  GREATER.  —  GENERAL  DESCRIPTION 
OP  A  COMET. — ^  COMETS  WITHOUT  TAILS,  OR  WITH  MORE  THAN 
ONE.  —  THEIR  EXTREME  .TENUITY.  —  THEIR   PROBABLE   STRUCTURE. 

—  MOTIONS  CONFORMABLE  TO  THE  LAW  OF  GRAVITY. — ACTUAL 
DIMENSIONS  OF  COMETS.  —  PERIODICAL  RETURN  OF  SEVERAL.  — 
HALLEY'S   COMET. — OTHER  ANCIENT  COMETS  PROBABLY  PERIODIC. 

— encke's  comet. — biela's.  —  faye's. — lexell's. — DE  vico's. 

— BRORSEN'S. — PETERS'S. — GREAT  COMET  OF  1843. — ITS  PROBABLE 
IDENTITY  WITH  SEVERAL  OLDER  COMETS.  —  GREAT  INTEREST  AT 
PRESENT   ATTACHED  TO  COMET ARY  ASTRONOMY,  AND  ITS  REASONS. 

—  REMARKS  ON  COMETARY   ORBITS  IN   GENERAL. 


f.  I-  ^ 


i 


(554.)  The  extraordinary  aspect  of  comets,  their  rapid  and  seemingly 
irregular  motions,  the  unexpected  manner  in  which  they  often  burst  upon 
us,  and  the  imposing  magnitudes  which  they  occasionally  assume,  have  ia 
all  ages  rendered  them  objects  of  astonishment,  not  unmixed  with  super- 
stitious  dread  to  the  uninstructed,  and  an  enigma  to  those  most  conversant 
with  the  wonders  of  creation  and  the  operations  of  natural  causes.  Even 
now,  that  we  have  ceased  to  regard  their  movements  as  irregular,  or  as 
governed  by  other  laws  than  those  which  retain  the  planets  in  their  orbits, 
their  intimate  nature,  and  the  offices  they  perform  in  the  economy  of  our 
system,  are  as  much  unknown  as  ever.  No  distinct  and  satisfactory 
account  has  yet  been  rendered  of  those  immensely  voluminous  append- 
ages  which  they  bear  about  with  them,  and  which  are  known  by  the  name 
of  their  tails,  (though  improperly,  since  they  often  precede  them  in  their 
motions,)  any  more  than  of  several  other  singularities  which  they 
present.  _  f 

(555.)  The  number  of  comets  which  have  been  astronomically  observed, 
or  of  which  notices  have  been  recorded  in  history,  is  very  great,  amount- 


296 


OUTLINES   OF  ASTRONOMY. 


ing  to  several  hundreds ; '  and  when  we  consider  that  in  the  earlier  ages 
of  astronomy,  and  indeed  in  more  recent  times,  before  the  invention  of 
the  telescope,  ociy  large  and  conspicuous  ones  were  noticed;  and  that 
since  due  attention  has  been  paid  to  the  subject,  scarcely  a  year  has  passed 
without  the  observation  of  one  or  two  of  these  bodies,  and  that  soiuetimes 
two  and  even  three  have  appeared  at  once ',  it  will  be  easily  supposed  that 
their  actual  number  muijt  be  at  least  many  thousands.  Multitudes, 
indeed,  must  escape  all  observation,  by  reason  of  their  paths  traversing 
only  that  part  of  the  heavens  which  is  above  the  horizon  in  the  daytime. 
Comets  so  circumstanced  can  jonly  become  visible  by  the  rare  coincidence 
of  a  total  eclipse  of  the  sun, — a  coincidence  which  happened,  as  related 
by  SenecK,  sixty-two  years  before  Christ,  when  a  large  comet  was  actually 
observed  very  near  the  sun.  Several,  however,  stand  on  record  as  having 
been  bright  enough  to  be  seen  with  the  naked  eye  in  the  daytime,  even 
at  noon  and  in  bright  sunshine.  Such  were  the  comets  of  1402,  1582, 
and  1843,  and  that  of  43  b.  c.  which  appeared  during  the  games  cele- 
brated by  Augustus  in  honour  of  Venus  shortly  after  the  death  of  Ga&sar, 
and  which  the  flattery  of  poets  declared  to  be  the  soul  of  that  hero  taking 
its  place  among  the  divinities. 

(556.)  That  feelings  of  awe  and  astonishment  should  be  excited  by  the 
sudden  and  unexpected  appearance  of  a  great  comet,  is  no  way  surprising; 
being,  in  fact,  according  to  the  accounts  we  have  of  such  events,  one  of 
the  most  imposing  of  all  natural  phenomena.  Comets  consist  for  the  most 
part  of  a  large  and  more  or  less  splendid,  but  ill-defined  nebulous  mass 
of  light,  called  the  head,  which  is  usually  much  brighter  towards  its 
centre,  and  offers  the  appearance  of  a  vivid  nucleus,  like  a  star  or  planet. 
From  the  head,  and  in  a  direction  ojyposite  to  that  in  which  the  sun  is 
situated  from  the  comet,  appear  to  diverge  two  streams  of  light,  which 
grow  broader  and  more  diffused  at  a  distance  from  the  head,  and  which 
most  commonly  close  in  and  unite  at  a  little  distance  behind  it,  but  some- 
times continue  distinct  for  a  great  part  of  their  course ;  producing  an  efiect 
like  that  of  the  trains  left  by  some  bright  meteors,  or  like  the  diverging 
fire  of  a  sky-rocket  (only  without  sparks  or  perceptible  motion.)     This  is 


SI;' 


m 


'  See  catalogues  in  the  Almagest  of  Riccioli ;  Pingre's  Cometographie  )  Delambre's 
Aafron.  vol.  iii. ;  Astronomische  Abhandlungen,  No.  1,  (which  contains  the  elements 
of  all  the  orbits  of  comets  which  have  been  computed  to  the  time  of  its  publication. 
1823;)  also  a  catalogue,  by  the  Rev.  T.  J.  Hussey.  Lond.  &  Ed.  Phil.  Mag.  vol.  ii. 
No.  9,  et  teq.  In  a  list  cited  by  Lalande  from  the  1st  vol.  of  the  Tables  de  Berlin, 
700  comets  are  enumerated.  See  also  notices  of  the  Astronomical  Society  and  Astron, 
Nachr.  passim.  A  great  many  of  the  more  ancient  comets  are  recorded  in  the  Chinese 
Annals,  and  in  some  cases  with  sufficient  precision  to  allow  of  the  calculation  of 
rudely  approximato  orbits  from  their  motions  so  described. 


EXTREME  TENUITT  OF  COMETS. 


297 


the  tail.  This  magnificent  appendage  attains  occasionally  an  immense 
appnrt^nt  length.  Aristotle  relates  of  the  tail  of  the  comet  of  371  b.  C, 
that  it  occupied  a  third  of  the  hemisphere,  or  60° ;  that  of  A.  D.  1618  is 
stated  to  have  been  attended  by  a  train  no  less  than  104 "^  in  length.  The 
comet  of  1680,  the  most  celebrated  of  modern  times,  and  on  many  ac* 
counts  the  most  remarkable  of  all,  with  a  head  not  exceeding  in  bright- 
ness a  star  of  the  second  magnitude,  covered  with  its  tail  an  extent  of 
more  than  70"  of  the  heavens,  or,  as  some  accounts  state,  90°  j  that  of 
the  comet  of  1769  extended  97°,  and  that  of  the  last  great  comet  (1843) 
was  estimated  at  about  65°  when  longest.  The  figure  {Jig,  ^,  Plate  II.) 
is  a  representation  of  the  comet  of  1819  —  by  no  means  one  of  the  most 
considerable,  but  which  was,  however,  very  conspicuous  to  the  naked  eye. 
(557.)  The  tail  is,  however,  by  no  means  an  invariable  appendage  of 
comets.  Many  of  the  brightest  have  been  observed  to  have  short  and 
feeble  tails,  and  a  few  great  comets  have  been  entirely  without  them. 
Those  of  1585  and  1763  oflFered  no  vestige  of  a  tail;  and  Cassini  describes 
the  comets  of  1665  and  1682  as  being  as  round'  and  as  well  defined  as 
Jupiter.  On  the  other  hand,  instances  are  not  wanting  of  comets  fur- 
nished with  many  tails  or  streams  of  diverging  light.  That  of  1744  had 
no  less  than  six,  spread  out  like  an  immense  fan,  extending  to  a  distance 
of  nearly  30°  in  length.  The  small  comet  of  1823  had  two,  making  an 
angle  of  about  160°,  the  brighter  turned  as  usual  from  the  sun,  the  fainter 
towards  it,  or  nearly  so.  The  tails  of  comets,  too,  are  often  somewhat 
curved,  bending,  in  general,  towards  the  region  which  the  comet  has  left, 
as  if  moving  somewhat  more  slowly,  or  as  if  resisted  in  their  course. 

(558.)  The  smaller  comets,  such  as  are  visible  only  in  telescopes,  or 
with  difficulty  by  the  naked  eye,  and  which  are  by  far  the  most  numerous, 
offer  very  frequently  no  appearance  of  a  tail,  and  appear  only  as  round 
or  somewhat  oval  vaporous  masses,  more  dense  towards  the  centre,  where, 
however,  they  appear  to  have  no  distinct  nucleus,  or  anything  which  seems 
entitled  to  be  considered  as  a  solid  body.  Stars  of  the  smallest  magni- 
tudes remain  distinctly  visible,  though  covered  by  what  appears  to  be  the 
densest  portion  of  their  substance;  although  the  same  stars  would  be 
completely  obliterated  by  a  moderate  fog,  extending  only  a  few  yards  from 
the  surface  of  the  earth.   And  since  it  is  an  observed  fact,  that  even  those 


{•fl 


Tl 


ni 


« 


mm 


'  This  description  however  applies  to  the  "  disc"  of  the  head  of  these  comets  as  seen 
in  a  telescope.  Cassini's  expressions  are,  "aussi  rond,  aussi  net,  et  aussi  clair  que 
Jupiter,"  (where  it  is  to  be  observed  tiiat  the  latter  epithet  must  by  no  means  be  trans- 
lated bright).  To  understand  this  passage  fully,  the  reader  must  refer  to  the  descrip- 
tion given  further  on,  of  the  "disc"  of  Haliey's  comet,  after  its  perihelio'i  passajD:^ 
m  1835-6. 


298 


OUTLINES   OP  ASTRONOMY. 


larger  comets  which  have  presented  the  appearance  of  a  nucleus  have  yet 
exhibited  no  phases,  though  we  ciuioot  doubt  that  they  shine  by  the  re- 
flected solar  light,  it  follows  that  even  these  can  only  be  regarded  as  great 
masses  of  thin  vapour,  susceptible  of  being  penetrated  through  their 
whole  substance  by  the  sunbeams,  and  reflecting  them  alike  from  their 
interior  parts  and  from  their  surfaces.  Nor  will  any  one  regard  this  ex- 
planation as  forced,  or  feel  disposed  to  resort  to  a  phosphorescent  quality 
in  the  comet  itself,  to  account  for  the  phenomena  in  question,  when  we 
consider,  (what  will  hereafter  be  shown)  the  enormous  magnitude  of  the 
space  thus  illuminated,  and  the  extremely  small  mats  which  there  is 
ground  to  attribute  to  these  bodiesit.  It  will  then  be  evident  that  the  most 
unsubstantial  clouds  which  float  in  the  highest  regions  of  our  atmosphere, 
and  seem  at  sunset  to  be  drenched  in  light,  and  to  glow  throughout  their 
whole  depth  as  if  in  actual  ignition,  without  any  shadow  or  dark  side, 
must  be  looked  upon  as  dense  and  massive  bodies  compared  with  the  filmy 
nnd  all  but  spiritual  texture  of  a  comet.  Accordingly,  whenever  p..;7erful 
telescopes  have  been  turned  on  these  bodies,  they  have  not  failed  to  dispel 
the  illusion  which  attributes  solidity  to  that  more  condensed  part  of  the 
bead,  which  appears  to  the  naked  eye  as  a  nucleus ;  though  it  is  true  that 
iu  some,  a  very  minute  stellar  point  hai  been  seen,  indicating  the  existence 
of  a  solid  body. 

(550.)  It  is  in  all  probability  to  the  feeble  coercion  of  the  elastic  power 
of  their  gaseous  parts,  by  the  gravitation  of  so  small  a  central  mass,  that 
we  must  attribute  this  extraordinary  development  of  the  atmospheres  of 
comets  If  the  earth,  retaining  its  present  size,  were  reduced,  by  any  in- 
ternaA  unange  (as  by  hollowing  out  its  central  parts)  to  one  thousandth  part 
of  its  actual  mass,  its  coercive  power  over  the  atmosphere  would  be  dimi- 
nished in  the  same  proportion,  and  in  consequence  the  latter  would  expand 
to  a  thousand  times  its  actual  bulk ;  and  indeed  much  more,  owing  to  the 
still  farther  diminution  of  gravity,  by  the  recess  of  the  upper  parts  from 
the  centre.'  An  atmosphere,  however,  free  to  expand  equally  in  all  di- 
rections, would  envelope  the  nucleus  spherically,  so  that  it  becomes  neces- 
sary to  admit  the  action  of  other  causes  to  account  for  its  enormous  exten- 
sion in  the  direction  of  the  tail,  —  a  subject  to  which  we  shall  presently 
take  occasion  to  recur. 

(560.^  That  the  luminous  part  of  a  comet  is  something  in  the  nature  of 


'  Newton  has  calculated  (Princ.  III.  p.  512,)  that  a  globe  of  air  of  ordinary  density 
at  the  earth's  surface,  of  one  inch  in  diameter,  if  reduced  to  the  density  due  to  the 
altitude  above  the  surface  of  one  radius  of  the  earth,  would  occupy  a  sphere  exceeding 
in  radius  the  orbit  of  Saturn.  The  tail  of  a  great  comet  then,  for  anght  we  can  tell, 
may  consist  of  only  a  very  few  pounds  or  even  ounces  of  matter. 


MOTIONS  OF  COMSrS. 


299 


a  emoke,  fog,  or  oload,  suspended  in  a  transparent  atmosphere,  is  evident 
from  a  fact  which  has  been  often  noticed,  viz. — that  the  portion  of  the  tail 
where  it  comes  np,  and  surrounds  the  head,  is  yet  separate  from  it  by  an 
interval  less  luminous,  as  if  sustained  and  kept  off  from  contact  by  a 
transparent  stratum,  as  we  often  see  one  layer  of  clouds  over  another  with 
a  considerable  clear  space  between.  These  and  most  of  the  other  facts  ob- 
served in  the  history  of  comets,  appear  to  indicate  that  the  structure  of  a 
comet,  as  seen  in  section  in  the  direction  of  its  length,  must  be  that  of  a 
hollow  envelope,  of  a  parabolic  form,  enclosing  near  its  vertex  the  nucleus 
and  head,  something  as  represented  in  the  annexed  figure.  This  would 
account  for  the  apparent  division  of  the  tail  into  two  principal  lateral 

Fig.  76. 


•  •!••••••••••••■■•«••• 


PI 


X 

111" 


branches,  the  envelope  being  oblique  to  the  line  of  sight  at  its  borders, 
and  therefore  a  greater  depth  of  illuminated  matte?  being  there  exposed 
to  the  eye.  In  all  probability,  however,  they  admit  great  varieties  of 
structure,  and  among  them  may  very  possibly  be  bodies  of  widely  diffe- 
rent physical  constitution,  and  there  is  no  doubt  that  one  and  the  snrae 
comet  at  different  epochs  undergoes  great  changes,  both  in  the  disposition 
of  its  materials  and  in  their  physical  state. 

(561.)  We  come  now  to  speak  of  the  motions  of  comets.  These  are 
apparently  most  irregular  and  capricious.  Sometimes  they  remain  in 
sight  only  for  a  few  days,  at  others  for  many  months ;  some  move  with 
extreme  slowness,  others  with  extraordinary  velocity;  while  not  unfre- 
quently,  the  two  extremes  of  apparent  speed  are  exhibited  by  the  same 
comet  in  different  parts  of  its  course.  The  comet  of  1472  described  an 
arc  of  the  heavens  of  40"  of  a  great  circle*  in  a  single  day.  Some 
pursue  a  direct,  some  a  retrograde,  and  others  a  tortuous  and  very  irregu- 
lar course ;  nor  do  they  confine  themselves,  like  the  planets,  within  any 
certain  region  of  the  heavens,  but  traverse  indifferently  every  part.  Their 
variations  in  apparent  size,  during  the  time  they  continue  visible,  are  no 
less  remarkable  than  those  of  their  velocity ;  sometimes  they  make  their 
first  appearance  as  faint  and  slow  moving  objects,  with  little  or  no  tail ; 


'  120°  in  extent  in  the  former  editions.    But  this  was  the  arc  described  tn  longitudo, 
and  the  cemet  at  the  time  referred  to  had  great  north  latitude. 


800 


OUTLINES   OP  ASTRONOMY. 


but  by  degrees  accelerate,  enlarge,  and  throw  out  from  them  this  appen- 
dage, which  increases  in  length  and  brightness  till  (as  always  happens  ia 
such  cases)  they  approach  the  sun,  and  are  lost  in  his  beams.  After  a 
time  they  again  emerge,  on  the  other  side,  receding  from  the  sun  with  a 
velocity  at  first  rapid,  but  gradually  decaying.  It  is  for  the  most  part 
after  thus  passing  the  sun,  that  they  shine  forth  in  all  their  splendour, 
and  that  their  tails  acquire  their  greatest  length  and  developemcnt ;  thutf 
indicating  plainly  the  action  of  the  sun's  rays  as  the  exciting  cause  of  that 
extraordinary  emanation.  As  they  continue  to  recede  from  the  sun,  their 
motion  diminishes  and  the  tail  dies  away,  or  is  absorbed  into  the  head, 
which  itself  grows  continually  feebler,  and  is  at  length  altogether  lost 
Bight  of,  in  by  far  the  greater  number  of  cases  never  to  be  seen  more. 

(562.)  Without  the  clue  furnished  by  the  theory  of  gravitation,  the 
enigma  of  these  seemingly  irregular  and  capricious  movements  might 
have  remained  for  ever  unresolved.  But  Newton,  having  demonstrated 
the  possibility  of  any  conio  section  whatever  being  described  about  the 
sun,  by  a  body  revolving  under  the  dominion  of  that  law,  immedititoly 
perceived  the  applicability  of  the  general  proposition  to  the  case  of  come- 
tary  orbits;  and  the  great  comet  of  1680,  one  of  the  most  remarkable  on 
reoord,  both  for  the  immense  length  of  its  tail  and  for  the  excessive  close- 
ness of  its  approach  to  the  sun  (within  one-sixth  of  the  diameter  of  that 
luminary),  afforded  him  an  excellent  opportunity  for  the  trial  of  bis 
theory.  The  success  of  the  attempt  was  complete.  He  ascertained  that 
this  comet  described  about  the  sun  as  its  focus  an  elliptic  orbit  of  so  great 
an  excentricity  as  to  be  undistinguishable  from  a  parabola,  (which  is  the 
extreme,  or  limiting  form  of  the  ellipse  when  the  axis  becomes  infinite,) 
and  that  in  this  orbit  the  areas  described  about  the  sun  were,  as  in  the 
planetary  ellipses,  proportional  to  the  times.  The  representation  of  the 
apparent  motions  of  this  comet  by  such  an  orbit,  throughout  its  whole 
observed  course,  was  found  to  be  as  satisfactory  as  those  of  the  motions 
of  the  planets  in  their  nearly  circular  paths.  From  that  time  it  became 
a  received  truth,  that  the  motions  of  comets  are  regulated  by  the  same 
general  laws  as  those  of  the  planets — the  difference  of  the  cases  consisting 
only  in  the  extravagant  elongation  of  their  ellipses,  and  in  the  absence 
of  any  limit  to  the  inclinations  of  their  planes  to  that  of  the  ecliptic — or 
any  general  coincidence  in  the  direction  of  their  motions  from  west  to 
east,  rather  than  from  east  to  west,  like  what  is  observed  among  the 
planets. 

(563.)  It  is  a  problem  of  pure  geometry,  from  the  general  laws  of 
elliptic  or  parabolic  motion,  to  find  the  situation  and  dimensions  of  the 
ellipse  or  parabola  which  shall  represent  the  motion  of  any  given  comet. 


MOTIONS  OF  COMETS. 


801 


Iq  general,  three  oomplete  observations  of  its  right  ascension  and  declina- 
tion, with  the  times  at  whiqh  they  were  mode,  suffice  for  the  solution  of 
this  problem,  (which  is,  however,  by  no  means  an  easy  one,)  and  for  the 
determination  of  the  elements  of  the  orbit.  These  consist,  mutatU  niMr 
tandis,  of  the  same  data  as  are  required  for  the  computation  of  the  mo- 
tion of  a  planet;  (that  is  to  say,  the  longitude  of  the  perihelion,  that  of 
the  ascending  node,  the  inclination  to  the  ecliptic,'  the  scmiaxis,  excen- 
tricity,  and  time  of  perihelion  passage,  as  also  whether  the  motion  is 
direct  or  retrograde ;)  and,  once  determined,  it  becomes  very  easy  to  com- 
pare them  with  the  whole  observed  course  of  the  comet,  by  a  process 
exactly  similar  to  that  of  art.  602,  and  thus  at  once  to  ascertain  their 
correctness,  and  to  put  to  the  severest  trial  the  truth  of  those  general 
laws  on  which  all  such  calculations  are  founded. 

(564.)  For  the  most  part,  it  is  found  that  the  motions  of  comets  may 
be  sufficiently  well  represented  by  parabolic  orbits,  —  that  is  to  say, 
ellipses  whose  axes  are  of  infinite  length,  or,  at  least,  so  very  long  that 
DO  appreciable  error  in  the  calculation  of  their  motions,  during  all  the 
time  they  continue  visible,  would  be  incurred  by  supposing  them  actually 
infinite.  The  parabola  is  that  conic  section  which  is  the  limit  between 
the  ellipse  on  the  one  hand,  which  returns  into  itself,  and  the  hyperbola 
on  the  other,  which  runs  out  to  infinity.  A  comet,  therefore,  which 
should  describe  an  elliptic  path,  however  long  its  axis,  must  have  visited 
the  sun  before,  and  must  again  return  (unless  disturbed)  in  some  dcter- 
minate  period,  —  but  should  its  orbit  be  of  the  hyperbolic  character,  when 
once  it  had  passed  its  perihelion,  it  could  never  more  return  within  the 
sphere  of  our  observation)  but  must  run  off  to  visit  other  systems,  or  be 
lost  in  the  immensity  of  space.  A  very  few  comets  have  been  ascertained 
to  move  in  hyperbolas',  but  many  more  in  ellipses.  These  latter,  in  so 
far  as  their  orbits  can  remain  unaltered  by  the  attractions  of  the  planets, 
must  be  regarded  as  permanent  members  of  our  system. 

(505.)  We  must  now  say  a  few  words  on  the  actual  dimensions  of 
comets.  The  calculation  of  the  diameters  of  their  heads,  and  the  lengths 
and  breadths  of  their  tails,  offers  not  the  slightest  difficulty  when  once 
the  elements  of  their  orbits  are  known,  for  by  these  we  know  their  real 
distances  from  the  earth  at  any  time,  and  the  true  direction  of  the  tail, 
which  we  see  only  foreshortened.  Now  calculations  instituted  on  these 
principles  lead  to  the  surprising  fact,  that  the  comets  are  by  far  the  most 
voluminous  bodies  in  our  system.  The  following  are  the  dimensions  of 
some  of  those  which  have  been  made  the  subjects  of  such  inquiry. 

'For  example,  that  of  1723,  calculated  by  Burckhardt ;  that  of  1771,  by  both  Burck 
hardt  and  Encke ;  and  the  second  comet  of  1818,  by  Rosenberg  and  Schwabs. 


802 


OUTLINES  OF  ASTRONOMY. 


(566.)  The  tail  of  the  great  oomet  of  1680,  immediately  after  its  peri< 
helioQ  passage,  was  found  by  Newton  to  have  no  lean  than  20000000  of 
leagues  in  length,  and  to  have  occupied  only  two  days  in  its  emission 
from  the  comet's  body  1  a  decisive  proof  this  of  its  being  darted  forth  by 
some  active  force,  the  origin  of  which,  to  judge  from  the  direction  of  lii^ 
tail,  must  be  sought  in  the  sun  itself.  Its  greatest  length  amounted  to 
41000000  leagues,  a  length  much  exceeding  the  whole  interval  between 
the  sun  and  earth.  The  tail  of  the  comet  of  1769  extended  16000000 
leagues,  and  that  of  the  great  comet  of  1811,  86000000.  The  portioti 
of  the  head  of  this  last,  comprised  within  the  transparent  atmospheric  en- 
velope which  separated  it  from  the  tail,  was  180000  leagues  in  diameter. 
It  is  hardly  conceivable,  that  matter  once  projected  to  such  enormous  dis- 
tances should  ever  be  collected  again  by  the  feeble  attraction  of  such  a 
body  as  a  comet — a  consideration  which  accounts  for  the  surmised  pro- 
gressive diminution  of  the  tails  of  such  as  have  been  frequently  observed. 

(567.)  The  most  remarkable  of  those  comets  which  have  been  ascer- 
tained to  move  in  elliptic  orbits  is  that  of  Halley,  so  called  from  the  Ic- 
brated  Edmund  Hallcy,  who,  on  calculating  its  elements  from  its  perihelion 
passage  in  1682,  when  it  appeared  in  great  splendour,  with  a  tail  30°  in 
length,  wa.s  led  to  conclude  its  identity  with  the  great  comets  of  1531  and 
1607,  whose  elements  he  had  also  ascertained.  The  intervals  of  these 
successive  apparitions  being  75  and  76  years,  Halloy  was  encouraged  to 
predict  its  reappearance  about  the  yv.  1759.  So  remarkable  a  predic- 
tion could  not  fail  to  attract  the  ottrt\tion  of  all  astronomers,  and,  as  the 
time  approached,  it  became  extremely  interesting  to  know  whether  the 
attractions  of  the  larger  planets  might  not  materially  interfere  with  it? 
orbitual  motion.  The  computation  of  their  influence  from  the  Newtonian 
law  of  gravity,  a  most  diflBcult  and  intricate  piece  of  calculation,  was 
undertaken  and  accomplished  by  Clairaut,  who  found  that  the  action  of 
Satun  would  retard  its  return  by  100  days,  and  that  of  Jupiter  by  ne 
less  than  518,  making  in  all  618  days,  by  which  the  expected  return 
would  happen  later  than  on  the  supposition  of  its  retaining'  un  iinaltered 
period, — and  that,  in  short,  the  time  of  the  expected  peribflicu  paj««»e 
would  take  ^.lace  within  a  month,  one  way  or  other,  of  :!.      of 

April,  1759. — It  actually  happened  on  the  12th  of  March  in  that  year 
Its  next  return  was  calculated  by  several  eminent  geometers',  and  fixed 
succesrvelj  for  the  4th,  the  7th,  the  11th,  and  the  26th  of  November, 
1835;  the  -""ro  klter  determinations  appearing  entitled  to  the  higher  de- 
gree of  con&^e>  ."1,  owing  partly  to  the  more  complete  discussion  bestowed 
on  the  opiiervaUons  'a.  .'682  aa^  1759,  and  partly  to  the  continually  im- 

*  Daii.'O'Mau,  Pontec^ulant,  Rosenberger,  and  Lehmana. 


proving  ''tate 
turbing     lect 
of  M.  Tiohma 
August  the  00 
as  an  cxceedii 
predicted  by 
August  it  be< 
latud  path  amo 
after  which,  v 
thou.'V  jk  <  ni 

♦  iJOUjjil' ui 

5-1,  of  Hay. 

(5bo.;  Altl 

apparition  was 

lively  sonsatio 

object  of  the  1 

mers,  furnishe 

bad  been  appl 

of  the  greater 

its  physical  s' 

sented  when  s( 

history.     Its 

that  of  a  snoal! 

having  a  mini 

within  it.     It 

developed,  an( 

or  5°  long  on 

20°)  on  the  1 

at  its  perihelic 

it  was  only  3* 

reason  to  beli( 

appe.ved,  as, 

very  day  of  i 

tail  being  the 

(509.)  By 

part  of  its  cai 

growth  of  the 

that  appenda] 

(the  very  day 

which  had  be 

much  bright 


BALLBY  S  COMET. 


808 


i'l 


proving  «tate  of  our  knowledge  of  the  inothdds  of  eBtimating  .^ho  dis- 
turbing leot  of  the  Bovoral  planctH.  The  last  <.f  tfioge  prcdiotiona,  that 
of  M.  Tiohmana,  was  publi.shcd  on  thi  _  h  of  Jul;  On  the  6th  of 
August  the  oomet  first  became  visiblo  in  the  U'^  atoios]*  lere  of  Ruine 
as  an  exceedingly  faint  telescopic  nebuiu,  within  a  degrofi  «f  its  place  as 
predicted  by  M.  Roscnberger  for  that  day.  On  or  about  tlio  20th  of 
August  it  became  generally  visible,  and,  pursuing  very  nearly  its  c>alcu< 
latud  path  among  the  stars,  passed  its  pcriholion  on  the  lOt  )f  Nover'her; 
after  whirh,  '*  j  course  carrying  it  south,  it  ceased  to  be  visi  'e  in  Ev».  ope, 
ihou  -^  ji  I'jn  inued  to  be  conspicuously  so  in  the  southen.  'icmispticre 
thiou^ii'Ui  ^t'ljAiary,  March,  and  April,  1886,  disappearing  li.mlly  on  the 
6'*i  of  iHay. 

(5bo .;  Although  the  appcaraiice  of  this  celebrated  comet  at  '"a  last 
apparition  was  not  such  as  might  bo  reasonably  considered  likely  tu  xcito 
lively  soiiJiations  of  terror,  even  in  superstitious  ages,  yet,  having  been  nn 
libjoct  of  the  most  diligent  attention  in  all  parts  of  the  world  to  asrr(  lo- 
ujers,  furnished  with  telescopes  very  fur  surpassing  in  power  those  v  i  h 
bad  been  applied  to  it  at  its  former  appearance  in  1759,  and  indeed  to  ..ly 
of  the  greater  comets  on  record,  the  opportunity  thus  afforded  of  studying 
its  physical  structure,  and  the  extraordinary  phasnomena  which  it  pre- 
sented when  so  examined,  have  rendered  this  a  memorable  epoch  in  cometic 
history.  Its  first  oppearance,  while  yet  very  remote  from  the  sun,  was 
that  of  a  small  round  or  somewhat  oval  nebula,  quite  destitute  of  tail,  and 
having  a  minute  point  of  more  concentrated  light  excentrically  situated 
within  it.  It  was  not  before  the  2d  of  October  that  the  tail  began  to  be 
developed,  and  thenceforward  increased  pretty  rapidly,  being  already  4° 
or  5°  long  on  the  5th.  It  attained  its  greatest  apparent  length  (about 
20°)  on  the  15th  of  October.  From  that  time,  though  not  yet  arrived 
at  its  perihelion,  it  decreased  with  such  rapidity,  that  already  on  the  29th 
it  was  only  3**,  and  on  November  the  5th  2J°  in  length.  There  is  every 
reason  to  believe  that  before  the  perihelion,  the  tail  had  altogether  dis- 
appeared, as,  though  it  continued  to  be  observed  at  Pulkowa  up  to  the 
very  day  of  its  perihelion  passage,  no  mention  whatever  is  made  of  any 
tail  being  then  seen. 

(509.)  By  far  the  most  striking  phaenomena,  however,  observed  in  this 
part  of  its  career,  were  those  which,  commencing  simultaneously  with  the 
growth  of  the  tan,  connected  themselves  evidently  with  the  production  of 
that  appendage  and  ita  projection  from  the  head.  On  the  2d  of  October 
(the  very  day  of  the  first  observed  commencement  of  the  tail)  the  nucleus, 
which  had  bet^  faint  and  smnil,  was  observed  suddenly  to  have  become 
much  brighter,  and  to  bo  iu  the  act  of  throwing  out  a  jet  or  stream  of 


f 


ti 


\ 


--■jf   r--- 


■>r- 


304 


OUTLINES   OF  ASTRONOMY. 


light  from  its  anterior  part,  or  that  turned  towards  the  sun.  This  ejection 
after  ceasing  awhile  was  resumed,  and  with  much  greater  apparent  vie- 
lence,  on  the  8th,  and  continued,  with  occasional  intermittences,  so  long 
as  the  tail  itself  continued  visible.  Both  the  form  of  this  luminous  ejec- 
tion, and  the  direction  in  which  it  issued  from  the  nucleus,  meanwhile 
underwent  singular  and  capricious  alterations,  the  different  phases  suc- 
ceeding each  other  with  such  rapidity  that  on  no  two  successive  nights 
were  the  appearances  alike.  At  one  time  the  emitted  jet  was  single,  and 
confined  within  narrow  limits  of  divergence  from  the  nucleus.  At  others 
it  presented  a  fan-shaped  or  swallow-tailed  form,  analogous  to  that  of  a 
gas-flame  issuing  from  a  flattened  orifice  :  while  at  others  again  two,  three, 
or  even  more  jets  were  darted  forth  in  different  directions.'  (See  figures 
a,  b,  c,  d,  plate  I.,  fig.  4,  which  represent,  highly  magnified,  the  appear- 
ances of  the  nucleus  with  its  jets  of  light,  on  the  8th,  9th,  10th,  and  12th 
of  October,  and  in  which  the  direction  of  the  anterior  portion  of  the  head, 
or  that  fronting  the  sun,  is  supposed  alike  in  all,  viz.  towards  the  upper 
part  of  the  engraving.  In  these  representations  the  head  itself  is  omitted, 
the  scale  of  the  figures  not  permitting  its  introduction :  e  represents  the 
nucleus  and  head  as  seen  October  9th  on  a  less  scale.)  The  direction  of 
the  principal  jet  was  observed  meanwhile  to  oscillate  to  and  fro  on  either 
side  of  a  line  directed  to  the  sun  in  the  manner  of  a  compass-needle  when 
thrown  into  vibration  and  oscillating  about  a  mean  position,  the  change 
of  direction  being  conspicuous  even  from  hour  to  hour.  These  jets, 
though  very  bright  at  their  point  of  emanation  from  the  nucleus,  faded 
rapidly  away,  and  became  diffused  as  they  expanded  into  the  coma,  at  the 
same  time  curving  backwards  as  streams  of  steam  or  smoke  would  do,  if 
thrown  out  from  narrow  orifices,  more  or  less  obliquely  in  opposition  to  a 
powerful  wind,  against  which  they  were  unable  to  make  way,  and  ulti- 
mately yielding  to  its  force,  so  as  to  be  drifted  back  and  confounded  in  a 
vaporous  train,  following  the  general  direction  of  the  current.'* 

(570.)  Reflecting  on  these  phaenomena,  and  carefully  considering  the 
evidence  afforded  by  the  numerous  and  elaborately  executed  drawings 
which  have  been  placed  on  record  by  observers,  it  seems  impossible  to 
avoid  the  following  conclusions.     1st.  That  the  matter  of  the  nucleus  of 


*  See  the  exquisite  lithographic  representations  of  these  phenomena  by  Bessel. 
Astron.  Nachr .  No.  302,  and  the  fine  series  by  Schwabe  in  No.  297  of  that  collec- 
tion, as  also  the  magnificent  drawings  of  Struve,  from  which  our  figures  a,  b,  c,  d,  aie 
copies. 

"  On  this  point  Schwabe's  and  Bessel's  drawings  are  very  express  and  unequiv- 
ocal. Struve's  attention  seems  to  have  been  more  especially  directed  to  the  scrv'iny 
of  the  nucleus. 


halley's  comet. 


305 


a  comet  is  powerfully  excited  and  dilated  into  a  vaporous  state  by  the 
action  of  the  sun's  rays,  escaping  in  streams  and  jets  at  those  points  of  its 
surface  which  oppose  the  least  resistance,  and  in  all  probability  throwing 
that  surface  or  the  nucleus  itself  into  irregular  motions  by  its  reaction  in 
the  act  of  so  escaping,  and  thus  altering  its  direction. 

2dly.  That  this  process  chiefly  takes  place  in  that  portion  of  the 
nucleus  which  is  turned  towards  the  sun ;  the  vapour  escaping  chiefly  in 
that  direction. 

3dly.  That  when  so  emitted,  it  is  prevented  from  proceeding  in  the 
direction  originally  impressed  upon  it,  by  some  force  directed  from  the 
sun,  drifting  it  back  and  carrying  it  out  to  vast  distances  behind  tlie 
nucleus,  forming  the  tail  or  so  much  of  the  tail  as  can  be  considered  as 
consisting  of  material  substance. 

4thly.  That  this  force,  whatever  its  nature,  acts  unequally  on  the  ma- 
terials of  the  comet,  the  greater  portion  remaining  unvaporized,  and  a 
considerable  part  of  the  vapour  actually  produced,  remaining  in  its  neigh- 
bourhood, forming  the  head  and  coma. 

5thly.  That  the  force  thus  acting  on  the  materials  of  the  tail  cannot 
possibly  be  identical  with  the  ordinary  gravitation  of  matter,  being  centri- 
fugal or  repulsive,  as  respects  the  sun,  and  of  an  energy  very  far  exceeding 
the  gravitating  force  towards  that  luminary.  This  will  be  evident  if  we 
consider  the  enormous  velocity  with  which  the  matter  of  the  tail  is  carried 
backwards,  in  opposition  both  to  the  motion  which  it  had  as  part  of  the 
nucleus,  and  to  that  which  it  acquired  in  the  act  of  its  emission,  both 
which  motions  have  to  be  destroyed  in  the  first  instance,  before  any  move- 
ment in  the  contrary  direction  can  be  impressed. 

6thly,  That  unless  the  matter  of  the  tail  thus  repelled  from  the  sun  be 
retained  by  a  peculiar  and  highly  energetic  attraction  to  the  nucleus,  dif- 
fering fnjm  and  exceptional  to  the  ordinary  power  of  gravitation,  it  must 
leave  the  nucleus  altogether  j  being  in  effect  carried  far  beyond  the  coer- 
cive power  of  so  feeble  :i  gravitating  force  as  would  correspond  to  the 
minute  mass  of  the  nucleus ;  and  it  is  therefore  very  conceivable  that  a 
comet  may  lose,  at  every  approach  to  the  sun,  a  portion  of  that  peculiar 
matter,  whatever  it  be,  on  which  the  production  of  its  tail  depends,  the 
remainder  being  of  course  less  excitable  by  the  solar  action,  and  more 
impassive  to  his  rays,  and  therefore,  'pro  tanto,  more  nearly  approximating 
to  the  nature  of  the  planetary  bodies. 

(571.)  After  the  perihelion  passage,  the  comet  was  lost  sight  of  for 
upwards  of  two  months,  and  at  its  reappearance  (on  the  24th  of  January 
1836)  presented  itself  under  quite  a  difibrent  aspect,  having  in  the  in- 
terval evidently  undergone  some  great  physical  change  which  had  operated 
20 


'■"-'W^. 


306 


OUTLINES  OP  ASTRONOM' 


an  entire  transformation  in  its  appearance.  It  no  longer  presented  any 
vestige  of  tail,  but  appeared  to  the  naked  eye  as  a  hazy  star  of  about  the 
fourth  or  fifth  magnitude,  and  in  powerful  telescopes  as  a  small,  round, 
well-defined  disc,  rather  more  than  2'  in  diameter,  surrounded  with  a 
nebulous  chevelure  or  coma  of  much  greater  extent.  Within  the  disc, 
and  somewhat  ezcentrically  situated,  a  minute  but  bright  nucleus  appeared, 
from  which  extended  towards  the  posterior  edge  of  the  disc  (or  that  remote 
from  the  sun)  a  short  vivid  luminous  ray.  (See  fig.  4  of  pi.  I.)  As  the 
comet  receded  from  the  sun,  the  coma  speedily  disappeared,  as  if  absorbed 
into  the  disc,  which,  on  the  bther  hand,  increased  continually  in  dimen- 
sions, and  that  with  such  rapidity,  that  in  the  week  elapsed  from  January 
25th  to  February  1st,  (calculating  from  micrometrical  measures,  and  from 
the  known  distance  of  the  comet  from  the  earth  on  those  days)  the  actual 
volume  or  real  solid  content  of  the  illuminated  space  had  dilated  in  tho 
ratio  of  upwards  of  40  to  1.  And  so  it  continued  to  swell  out  with  un- 
diminished rapidity,  until  from  this  cause  alone  it  ceased  to  be  visible,  the 
illumination  becoming  fainter  as  the  magnitude  increased  j  till  at  fcngth 
the  outline  became  undistinguishable  from  simple  want  of  light  to  trace 
it.  While  this  increase  of  dimension  proceeded,  the  form  of  the  disc 
passed,  by  gradual  and  successive  additions  to  its  length  in  the  direction 
opposite  to  the  sun,  to  that  of  a  paraboloid,  as  represented  in  g,  fig.  4, 
plate  I.,  the  anterior  curved  portion  preserving  its  planetary  sharpness, 
but  the  base  being  faint  and  ill-defined.  It  is  evident  that  had  this  pro- 
cess continued  with  suflScient  light  to  render  the  result  visible,  a  tail  would 
have  been  ultimately  reproduced;  but  the  increase  of  dimension  being 
accompanied  with  diminution  of  brightness,  a  short,  imperfect,  and  as  it 
were  rudimentary  tail  only  was  formed,  visible  as  such  for  a  few  nights  to 
the  naked  eye,  or  in  a  low  magnifying  telescope,  and  that  only  when  the 
comet  itself  had  begun  to  fade  away  by  reason  of  its  increasing  distance. 
(572.)  While  the  parabolic  envelope  was  thus  continually  dilating  and 
growing  fainter,  the  nucleus  underwent  little  change,  but  the  ray  proceed- 
ing from  it  increased  in  length  and  comparative  brightness,  preserving  all 
the  time  its  direction  along  the  axis  of  the  paraboloid,  and  offering  none 
of  those  irregular  and  capricious  phaenomena  which  characterized  the  jets 
of  light  emitted  anteriorly,  previous  to  the  perihelion.  If  the  oflice  of 
those  jets  was  to  feed  the  tail,  the  converse  office  of  conducting  back  its 
successively  condensing  matter  to  the  nucleus  would  seem  to  be  that  of 
the  ray  now  in  question.  By  degrees  this  also  faded,  and  the  last  appear- 
ance presented  by  the  comet  was  that  which  it  offered  at  its  first  appear- 
ance in  August ;  viz.  that  of  a  small  round  nebula  with  a  bright  point  in 
or  near  the  centre. 


OTHER  PERIODICAL   COMETS. 


807 


(573.)  Besides  the  comet  of  Halley,  several  other  of  the  great  comets 
recorded  in  history  have  been  surmised  with  more  or  less  probability  to 
return  periodically,  and  therefore  to  move  in  elongated  ellipses  around  the 
sun.  Such  is  the  great  comet  of  1680,  whose  period  is  estimated  at  575 
years,  and  which  is  considered,  with  the  highest  appearance  of  probability, 
to  be  identical  with  a  magnificent  comet  observed  at  Constantinople  and 
in  Palestine,  and  referred  by  contemporary  historians,  both  European  and 
Chinese,  to  the  year  a.  d.  1105 ;  with  that  of  A.  D.  575,  which  was  seen 
at  noon-day  close  to  the  sun  j  with  the  comet  of  43  b.  c,  already  spoken 
of  as  having  appeared  after  the  death  of  Caesar,  and  which  was  also 
observed  in  the  day-time ;  and  finally  with  two  other  comets,  mention  of 
which  occurs  in  the  Sibylline  Oracles,  and  in  a  passage  of  Homer,  and 
which  are  referred,  as  well  as  the  obscurity  of  chronology  and  the  indica- 
tions themselves  will  allow,  to  the  years  618  and  1194  b.  c.  It  is  to  the 
assumed  near  approach  of  this  comet  to  the  earth  about  the  time  of  the 
Deluge,  that  Whiston  ascribed  that  overwhelming  tide  wave  to  whose 
agency  his  wild  fancy  ascribed  that  great  catastrophe — a  speculation,  it  is 
needless  to  remark,  purely  visionary. 

(574.)  Another  great  comet,  whose  return  in  the  year  actually  current 
(1848)  has  been  considered  by  more  than  one  eminent  authority  in  this 
department  of  astronomy'  highly  probable,  is  that  of  1556,  to  the  terror 
of  whose  aspect  some  historians  have  attributed  the  abdication  of  the 
Emperor  Charles  V.  This  comet  is  supposed  to  be  identical  with  that 
of  1264,  mentioned  by  many  historians  as  a  great  comet,  and  observed 
also  in  China,  —  the  conclusion  in  this  case  resting  upon  the  coincidence 
of  elements  calculated  on  the  observations,  such  as  they  are,  which  have 
been  recorded.  On  the  subject  of  this  coincidence  Mr.  Rind  has  recently 
entered  into  many  elaborate  calculations,  the  result  of  which  is  strongly 
in  favour  of  the  supposed  identity.  This  probability  is  farther  increased 
by  the  fact  of  a  comet  with  a  tail  of  40°  and  a  head  bright  enough  to  be 
visible  after  sunrise  having  appeared  in  A.  d.  975;  and  of  two  others 
having  been  recorded  by  the  Chinese  annalists  in  a.  d.  395  and  104.  It 
is  true  that  if  these  be  the  same,  the  mean  period  would  be  somewhat 
short  of  292  years.  But  the  eflFect  of  planetary  perturbation  might 
reconcile  even  greater  differences,  and  though  up  to  the  time  of  our 
writing  no  such  comet  has  yet  been  observed,  at  least  another  year  must 
elapse  before  its  return  can  be  pronounced  hopeless. 

(575.)  In  1661,  1532,  1402,  1145,  891,  and  243  great  comets 
appeared — that  of  1402  being  bright  enough  to  be  seen  at  noon-day.  A 
period  of  129  years  would  conciliate  all  these  appearances,  and  should 
'  Pingr^,  Cometographie,  i.  411.    Lalande,  Astr.  3185. 


n.a 


308 


OUTLINES   OF  ASTRO^'OMY. 


have  brought  back  the  comet  in  1789  'or  1790  (other  circumstances 
agreeing.)  That  no  such  comet  was  observed  about  that  time  is  no  proof 
that  it  did  not  return,  since,  owiag  to  the  situation  of  its  orbit,  had  the 
perihelion  passage  taken  place  in  July  it  might  have  escaped  observation. 
Mechain,  indeed,  from  an  elaborate  discussion  of  the  observations  of  1532 
and  1661,  came  to  the  conclusion  that  these  comets  were  not  the  same ; 
but  the  elements  assigned  by  Olbers  to  the  earlier  of  them,  differ  so 
widely  from  those  of  Mechain  for  the  same  comet  on  the  one  hand,  and 
agree  so  well  with  those  of  the  last  named  astronomer  for  the  other,' 
that  we  are  perhaps  justified  in  regarding  the  question  as  not  yet  set  at 
rest. 

(576.)  We  come  now,  however,  to  a  class  of  comets  of  short  period, 
respecting  whose  return  there  is  no  doubt,  inasmuch  as  two  at  least  of 
them  have  been  identified  as  having  performed  successive  revolutions 
round  the  sun;  have  had  their  return  predicted  already  several  times; 
and  have  on  each  occasion  scrupulously  kept  to  their  appointments.  The 
first  of  thesd  is  the  comet  of  Encke,  so  called  from  Professor  Encke  of 
Berlin,  who  first  ascertained  its  periodical  return.  It  revolves  in  an 
ellipse  of  great  excentricity  (though  not  comparable  to  that  of  Halley's,) 
the  plane  of  which  is  inclined  at  an  angle  of  about  13°  22'  to  the  plane 
of  the  ecliptic,  and  in  the  short  period  of  1211  days,  or  about  3J  years. 
This  remarkable  discovery  was  made  on  the  occasion  of  its  fourth  recorded 
appearance,  in  1819.  From  an  ellipse  then  calculated  by  Encke,  its 
return  in  1822  was  predicted  by  him,  and  observed  at  Paramatta,  in  New 
South  Wales,  by  M.  Riimker,  being  invisible  in  Europe :  since  which  it 
has  been  re-predicted  and  re-observed  in  all  the  principal  observatories, 
both  in  the  northern  and  southern  hemispheres,  as  a  phenomenon  of 
regular  occurrence. 

(577.)  On  comparing  the  intervals  between  the  successive  perihelion 
passages  of  this  comet,  after  allowing  in  the  most  careful  and  exact  manner 
for  all  the  disturbances  due  to  the  actions  of  the  planets,  a  very  singular 
fact  has  come  to  light,  viz.  that  the  periods  are  continually  diminishing, 
or,  in  other  words,  the  mean  distance  from  the  sun,  or  the  major  axis  of 
the  ellipse,  dwindling  by  slow  and  regular  degrees  at  the  rate  of  about 
O^ll  per  revolution.  This  is  evidently  the  efiect  which  would  be  pro- 
duced by  a  resistance  experienced  by  the  comet  from  a  very  rare  ethereal 
medium  pervading  the  regions  in  which  it  moves ;  for  such  resistance,  by 
diminishing  its  actual  velocity,  would  diminish  also  its  centrifugal  force, 
and  thus  give  the  sun  more  power  over  it  to  draw  it  nearer.  Accordingly 
this  is  the  solution  proposed  by  Encke,  and  at  present  generally  received. 
'  Se«  Schumacher's  Catal.  Astron.  Abhandl.  i. 


DOUBLE  COMET  OF  BIELA. 


869 


It  will,  therefore,  probably  fall  ultimately  into  the  sun,  should  it  not  first 
be  dissipated  altogether, — a  thing  no  way  improbable,  when  the  lightness 
of  its  materials  is  considered. 

(578.)  By  measuring  the  apparent  magnitude  of  this  comet  at  different 
distances  from  the  sun,  and  taence,  from  a  knowledge  of  its  actual  dis- 
tance from  the  earth  at  the  time,  concluding  its  real  volume,  it  has  been 
ascertained  to  contract  in  bulk  as  it  approaches  to,  and  to  expand  as  it 
recedes  f  -om,  that  luminary.  M.  Valz,  who  was  the  first  to  notice  this 
fact,  accounts  for  it  by  supposing  it  to  undergo  a  real  compression  or  con- 
densation of  volume  arising  from  the  pressure  of  an  aethereal  medium  which 
he  conceives  to  grow  more  dense  in  the  sun's  neighbourhood.  But  such 
an  hypothesis  is  evidently  inadmissible,  since  it  would  require  us  to  assume 
the  exterior  of  the  comet  to  be  in  the  nature  of  a  skin  or  bag  impervious 
to  the  compressing  medium.  The  phenomenon  is  analogous  to  the  increase 
of  dimension  above  described  as  observed  in  the  comet  of  Halley  when  in 
the  act  of  receding  from  the  sun,  and  is  doubtless  referable  to  a  similar 
cause,  viz.  the  alternate  conversion  of  evaporable  matter  into  the  states  of 
visible  cloud  and  invisible  gas  by  the  alternating  action  of  cold  and  heat. 
This  comet  has  no  tail,  but  offers  to  the  view  only  a  small  ill-defined 
nucleus,  excentrically  situated  within  a  more  or  less  elongated  oval  mass 
of  vapours,  being  nearest  to  that  vertex  which  is  towards  the  sun. 

(579.)  Another  comet  of  short  period  is  that  of  Biela,  so  called  from 
M.  Biela,  of  Josephstadt,  who  first  arrived  at  this  interesting  conclusion 
on  the  occasion  of  its  appearance  in  1826.  It  is  considered  to  be  identi- 
cal with  comets  which  appeared  in  1772, 1805,  &c.,  and  describes  its  very 
excentric  ellipse  about  the  sun  in  2410  days  or  about  6|  years;  and  in  a 
plane  inclined  12°  34'  to  the  ecliptic.  It  appeared  again  according  to  the 
prediction  in  1832,  and  in  1846.  Its  orbit,  by  a  remarkable  coincidence, 
very  nearly  intersects  that  of  the  earth ;  and  had  the  latter  at  the  time  of 
its  passage  in  1832,  been  a  month  in  advance  of  its  actual  place,  it  would 
have  passed  through  the  comet, — a  singular  rencontre,  perhaps  not  un- 
attended with  danger.' 


■■\( 


,  f 


\  • 


I  ' 


^.  m 


'pt 


*■  Should  calculation  establish  the  fact  of  a  resistance  experienced  also  by  this  comet, 
the  subject  of  periodical  comets  will  assume  an  extraordinary  degree  of  interest.  It 
cannot  be  doubted  that  many  more  will  be  discovered,  and  by  their  resistance  questions 
will  come  to  be  decided,  such  as  the  following : — What  is  the  law  of  density  of  the  re- 
sisting medium  which  surrounds  the  sun  ?  Is  it  at  rest  or  in  motion  ?  If  the  latter,  in 
what  direction  does  it  move  ?  Circularly  round  the  sun,  or  traversing  space  ?  If  cir- 
cularly, in  what  plane  ?  It  is  obvious  that  a  circular  or  vorticose  motion  of  the  ether 
would  accelerate  some  cornels  and  retard  others,  according  as  their  revolution  was,  rela- 
tive to  such  motion,  direct  or  retrograde.  Supposing  the  neighbourhood  of  the  sun  to 
be  filled  with  a  material  fluid,  it  is  not  conceivable  that  the  circulation  of  the  planets  in 


4    J* 


810 


OUTLINES  OP  ASTRONOMY. 


(580.)  This  comet  is  small  and  hardly  visible  to  the  naked  "iy*  even 
when  brightest.  Nevertheless,  as  if  to  make  up  for  its  seeming  iQen^jIfi. 
cance  by  the  interest  attaching  to  it  in  a  physical  point  of  view,  it  exhi- 
bited at  its  last  appearance,  in  1846,  a  phtopomenon  which  struck  every 
astronomer  with  amazement,  as  a  thing  without  previous  exampb  in  the 
history  of  our  system.'  It  was  actually  seen  to  separate  itself  into  two  dis- 
tinct comets,  which,  after  thus  parting  company,  continued  to  journey  along 
amicably  through  an  arc  of  upwards  of  70°  of  their  apparent  orbit,  keeping 
all  the  while  within  the  same  field  of  view  of  the  telescope  pointed  towards 
them.  The  first  indication  of  something  unusual  being  about  to  take 
place,  might  be,  perhaps,  referred  to  the  19th  of  December,  1845,  when 
the  comet  appeared  pear-shaped,  the  nebulosity  being  unduly  elongated  in 
the  north  following  direction.'  But  on  the  13th  of  January,  at  ^yash- 
ington  in  America,  and  on  the  15th  and  subsequently  in  every  part  of 
Europe,  it  was  distinctly  seen  to  have  become  double  j  a  very  small  and 
faint  cometic  body,  having  a  nucleus  of  its  own,  being  observed  appended 
to  it,  at  a  distance  of  about  2'  (in  arc)  from  its  centre,  and  in  a  diifection 
forming  an  angle  of  about  328°  with  the  meridian,  running  northwards 
from  the  principal  or  original  comet  (see  art.  204).  From  this  time  the 
separation  of  the  two  comets  went  on  progressively,  though  slowly.  On 
the  30th  of  January,  the  apparent  distance  of  the  nucleus  had  increased 
to  3',  on  the  7th  of  February  to  4',  and  on  the  13th  to  5',  and  so  on, 
until  on  the  5th  of  March  the  two  comets  were  separated  by  an  interval 
of  9'  19",  the  apparent  direction  of  the  line  of  junction  all  the  while 
varying  but  little  with  respect  to  the  parallel.' 

(581.)  During  this  separation  very  remarkable  changes  were  observed 
to  be  going  on  both  in  the  original  comet  and  its  companion.     Both  had 

it  for  ages  should  not  have  impressed  upcn  it  some  degree  of  rotation  in  their  own 
direction.  And  this  may  preserve  them  from  the  extreme  effects  of  accumulated 
resistance. — Author. 

*  Perhaps  not  quite  so.  To  say  nothing  of  a  singular  surmise  of  Kepler,  that  two 
great  comets  seen  at  once  in  1618,  might  be  a  single  comet  separated  into  two,  the  fol- 
lowing passage  of  Helvelius  cited  by  M.  Littrow  (Nachr.  564)  does  really  seem  to 
refer  to  some  pheenomenon  bearing  at  least  a  certain  analogy  to  it.  "  In  ipso  disco," 
he  says  (Cometographia,  p.  326)  "  quatuor  vel  quinque  corpuscula  quaedam  sive-nu- 
cleos  reliquo  corpore  aliquanto  densiores  ostendebat." 

'  According  to  Mr.  Hind's  observation.  But  there  can  be  little  doubt  that  by  a  mis- 
take of  the  most  common  occurrence,  when  no  measure  of  the  position  is  taken,  north 
ibllowing  is  an  error  of  entry  or  printing  for  north  ^.receding  (n  f  for  n  p).  In  fact,  an 
elongation  from  north  following  to  south  preceding  would  agree  with  the  regular  direc- 
tion of  the  tail  a  id  would  occasion  no  remark. 

*  By  tar  the  greater  portion  of  this  increase  of  apparent  distance  was  due  to  the 
comet's  increased  proximity  to  the  earth.  The  real  increase  reduced  to  a  distance  ^r  1 
of  the  comet  was  at  the  rate  of  about  3"  per  diem. 


DOUBLE  COMET  OF  BIELA. 


811 


nuclei,  both  had  Bhort  tails,  parallel  in  direction,  and  nearly  perpendicular 
to  the  line  of  junction,  but  whereas  at  its  first  observation  on  Janupry 
18th,  the  new  comet  was  extremely  small  and  faint  in  comparison  with 
the  old,  the  difference  both  in  point  of  light  and  apparent  magnitude  di- 
luiuisbed.  On  the  10th  of  February,  they  were  nearly  equal,  although 
the  day  before  the  moonlight  had  effaced  the  new  one,  leaving  th3  other 
bright  enough  to  be  well  observed.  On  the  14th  and  16th,  however,  the 
new  comet  had  gained  a  decided  superiority  of  light  over  the  old,  pre- 
senting at  the  same  time  a  sharp  and  starlike  nucleus,  compared  by  Lieut. 
Maury  to  a  diamond  spark.  But  this  state  of  things  was  not  to  continue. 
Already,  on  the  18th,  the  old  comet  had  regained  its  superiority,  being 
nearly  twice  as  bright  as  its  companion,  and  offering  an  unusually  bright 
and  starlike  nucleus.  From  this  period  the  new  companion  began  to  fade 
away,  but  continued  visible  up  to  the  15th  of  March.  On  the  24th  the 
comet  was  again  single,  and  on  the  22d  of  April  both  had  disappeared. 

(582.)  While  this  singular  interchange  of  light  was  going  forwards, 
indications  of  some  sort  of  communication  between  the  comets  were  exhi- 
bited. The  new  or  companion  comet,  besides  its  tail,  extending  in  a 
direction  parallel  to  that  of  the  other,  threw  out  a  faint  arc  of  light,  which 
extended  as  a  kind  of  bridge  from  the  one  to  the  other;  and  after  the 
restoration  of  the  original  comet  to  its  former  preeminence,  it,  on  its  part, 
threw  forth  additional  rays,  so  as  to  present  (on  the  22d  and  23d  Febru- 
ary) the  appearance  of  a  comet  with  three  faint  tails,  forming  angles  of 
about  120°  with  each  other,  one  of  which  extended  towards  its  com- 
panion.' 

(583.)  Professor  Plantamour,  director  of  the  observatory  of  Geneva, 
having  investigated  the  orbits  of  both  these  comets  as  separate  and  inde- 
pendent bodies,  from  the  extensive  and  careful  series  of  observations 
made  upon  them,  has  arrived  at  the  conclusion  that  the  increase  of  dis- 
tance between  the  two  nuclei,  at  least  during  the  interval  from  February 
10<A  to  March  22c?,  was  simply  apparent,  being  due  to  the  variation  of 
distance  from  the  earth,  and  to  the  angle  under  which  their  line  of  junc- 
tion presented  itself  to  the  visual  ray ;  the  real  distance  during  all  that 
interval  (neglecting  small  fractions)  having  been  on  an  average  about 
thirty-nine  times  the  semi-diameter  of  the  earth,  or  less  than  two-thirds 
the  distance  of  the  moon  from  its  centre.  From  this  it  would  appear, 
that  already,  at  this  distance,  the  two  bodies  had  ceased  to  exercise  any 

'  These  last-mentioned  particulars  rest  on  the  testiniony  of  Lieutenant  Maury  of 
Washington,  who  had  the  advantage  of  using  a  nine-inch  object-glass  of  Munich 
manufacture.    It  does  not  appear  that  any  large  telescope  was  turned  upon  it  in  Eu 
rope  on  the  dates  in  question. 


m 


812 


OUTLINES  OF  ASTRONOMY. 


percci)tihle  amount  of  perturbative  gravitation  on  each  other ;  as,  indeed, 
from  the  probable  minuteness  of  comctary  masses  ^e  might  reasonably 
expect.  Calculating  upon  the  elements  assigned  by  him',  wo  find  16''-4 
for  the  interval  of  their  next  perihelion  passages.  And  it  will  be,  there- 
fore, necessary  at  their  next  reappearance,  to  look  out  for  each  comet  as  i, 
separate  and  independent  body,  computing  its  place  from  these  elements 
as  if  the  other  had  no  existence.  Nevertheless,  as  it  is  still  perfectly 
possible  that  some  link  of  connection  may  subsist  between  them,  (if,  in- 
deed,  by  some  unknown  process  the  companion  has  not  been  actually 
reabsorbed,)  it  will  not  be  advisable  to  rely  on  this  calculation  to  the 
neglect  of  a  most  vigilant  search  throughout  the  whole  neighbourhood  of 
the  more  conspicuous  one,  lest  the  opportunity  should  be  lost  of  pursuing 
to  its  conclusion  the  history  of  this  strange  occurrence. 

(584.)  A  third  comot  of  short  period  has  still  more  recently  been  added 
to  our  list  by  M.  Faye,  of  the  observatory  of  Paris,  who  detected  it  on 
the  22d  of  November  1843.  A  very  few  observations  sufficed  to  show 
that  no  parabola  would  satisfy  the  conditions  of  its  motion,  and  tha'.  to 
represent  them  completely,  it  was  necessary  to  assign  to  it  an  elliptic  orbit 
of  very  moderate  excentricity.  The  calculations  of  M.  Nicolai,  subse- 
quently revised  and  slifi;htly  corrected  by  M.  Leverrier,  have  shown  that 
an  almost  perfect  representation  of  its  motions  during  the  whole  period 
of  its  visibility  would  be  affijirded  by  assuming  it  to  revolve  in  a  period  of 
27l7''-08  (or  somewhat  less  than  7i  years)  in  an  ellipse  whose  excen- 
tricity is  0-55596,  and  inclination  to  the  ecliptic  11°  22'  31";  and  taking 
this  for  a  basis  of  further  calculation,  and  by  means  of  these  data  and  the 
other  elements  of  the  orbit  estimating  the  effect  of  planetary  perturbation 
during  the  revolution  now  in  progress,  he  has  fixed  its  next  return  to  the 
perihelion  for  the  3d  of  April  1851,  with  a  probable  error  one  way  or 
other  not  exceeding  ono  or  two  days. 

(585.)  The  effect  of  pi;  notary  perturbation  on  the  motion  of  comets 
has  been  more  than  once  alluJed  to  in  what  has  been  above  said.  With- 
out going  minutely  into  this  parf.  of  the  subject,  which  will  be  better  un- 
derstood after  the  perusal  of  a  sv.bsequent  chapter,  it  must  be  obvious, 
'.hat  as  the  orbits  of  comets  are  very  excentric,  and  inclined  in  all  sorts  of 

1  Original  Comet.  Companion. 

Periheiion  pasaa-je,  \84'6,  Feb.  11-00476 11-07HI  Geneva  m.t. 

Long,  seriiia.fis  majp; 


Perihelion  disti.ncci 

Angle  of  exc;ntricity  or  whose 

sine^'e 


0-.')471002 0-5451271 

9  9327011 9-9326965 


Inclination 12 

Node  Si 245 

Perihelion 109 


49°  12'  2"-5 49°  6'  14"-4 


34  53  -3 12    34  14  -3 

54  38  -8 245   56     1-7 

2  20-1 109      2  39  -6 

Mean  equinox  of  1846,  '0. 


COMETS  OF  LEXBLL  AND  DE  YICO. 


818 


angles  to  the  ecliptic,  they  must  in  many  instances,  if  not  actually  inter 
sect,  at  least  pass  very  near  to  the  orbits  of  some  of  the  planets.  Wb 
have  already  seen,  for  instance,  that  the  orbit  of  Bicla's  comet  so  nearly 
iutersccts  that  of  the  earth,  that  an  actual  collision  is  not  impossible,  and 
indeed  (supposing  neither  orbit  variable)  must  in  all  likelihood  happen  in 
the  lapse  of  some  millions  of  years.  Neither  are  instances  wanting  of 
comets  having  actually  approached  the  earth  within  comparatively  short 
distances,  as  that  of  1770,  which  on  the  1st  of  July  of  that  year  was 
witin  little  more  than  seven  times  the  moon's  distance.  The  same  comet 
in  1767  passed  tTupiter  at  a  distance  only  one  dSth  of  the  radius  of  that 
planet's  orbit,  and  it  has  :ecn  rendered  extremely  probable  that  it  is  to 
the  disturbance  its  former  orbit  underwent  during  that  appulse  that  wo 
owe  its  appearance  within  our  own  range  of  vision.  This  exceedingly  re- 
markable comet  was  found  by  Lexell  to  describe  an  elliptic  orbit  wich  an 
excentricity  of  0*7858,  with  a  periodic  time  of  about  five  years  and  a 
half,  and  in  a  plane  only  1"  34'  inclined  to  the  ecliptic,  having  passed  its 
perihelion  on  the  13tj  f  August  1770.  Its  return  of  course  was  eagerly 
expected,  but  in  vain,  for  the  comet  has  never  been  seen  since.  Its  ob- 
servation on  its  first  return  in  1776  was  rendered  impossible  by  the  rela- 
tive situations  of  the  perihelion  and  of  the  earth  at  the  time,  and  before 
another  revolution  could  be  accomplished  (as  has  since  been  ascertained,) 
viz;  about  the  23d  of  August  1779,  by  a  singular  coincidence  it  again 
approached  Jupiter  within  one  491st  part  of  its  distance  from  the  sun, 
being  nearer  to  that  planet  by  one-fifth  than  its  fourth  satellite.  No 
wonder,  therefore,  that  the  planet's  attraction  (which  at  that  distance 
would  exceed  that  of  the  sun  in  the  proportion  of  at  least  200  to  1) 
should  completely  alter  the  orbit  and  deflect  it  into  a  curve,  not  one  of 
whose  elements  would  have  the  least  resemblance  to  those  of  the  ellipse 
of  Lexell.  It  is  worthy  of  notice  that  by  this  rencontre  with  the  system 
of  Jupiter's  satellites,  none  of  their  motions  suflFered  any  perceptible 
derangement,  —  a  sufficient  proof  of  the  smallness  of  its  mass.  Jupiter 
indeed,  seems,  by  some  strange  fatality,  to  be  constantly  in  the  way  of 
comets,  and  to  serve  as  a  perpetual  stumbling-block  to  them. 

(586.)  On  the  22nd  of  August,  1844,  Signor  De  Vico,  director  of  the 
observatory  of  the  CoUegio  Romano,  discovered  a  comet,  the  motions  of 
which,  a  very  few  observations  sufficed  to  show,  deviated  remarkably  from 
a  parabolic  orbit.  It  passed  its  perihelion  on  the  2nd  of  September,  and 
continued  to  be  observed  until  the  7th  of  December.  Elliptic  elements 
of  this  comet,  agreeing  remarkably  well  with  each  other,  were  accordingly 
calculated  by  several  astronomers ;  from  which  it  appears  that  the  period 
of  revolution  is  about  1990  days,  or  5  J  (5-4357)  years,  which  (supposing 


m 


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zu 


OUTUNBS   OF  i\8TR0N0MY. 


its  orbit  undisturbed  in  the  interim)  would  bring  it  back  to  the  pcrili  lion 
on  or  about  the  18th  of  January,  1850.  As  the  assemblage  and  com. 
parison  of  these  elements  thus  computed  independentlj,  will  servo  better, 
porhaps,  than  any  other  example,  to  afford  the  student  an  idea  of  tho 
degree  of  arithmetical  certainty  capable  of  being  attained  in  this  branch 
of  astronomy,  difficult  and  complex  as  the  calculations  themselves  arc,  and 
liable  to  error  as  individual  observations  of  a  body  so  ill-defined  us  the 
smaller  comets  are  for  the  most  part ;  we  shall  present  them  in  a  tabular 
form,  as  on  the  next  page :  the  elements  being  as  usual ;  the  time  of  pori- 
helion  passage,  longitude  of  the  perihelion,  that  of  the  ascending  node, 
the  inclination  to  the  ecliptic,  semiaxis  and  cxcentricity  of  the  orbit,  and 
the  periodic  time. 

This  comet,  when  brightest,  was  visible  to  the  naked  eye,  and  had  a 
small  tail.  It  is  especially  interesting  to  astronomers  from  the  circuin. 
stance  of  its  having  been  rendered  exceedingly  probable  by  the  rt  s  jarches 
of  M.  Leverrier,  that  it  is  identical  with  one  which  appeared  in  \()7S  witli 
some  of  its  elements  considerably  changed  by  perturbation.  Thia  comet 
is  further  remarkable,  from  having  been  concluded  by  Messrs.  Laugier 
and  Mauvais,  to  be  identical  with  the  comet  of  1585  observed  by  Tycho 
Brahe,  and  possibly  also  with  those  of  1748,  1766,  and  1819. 

(587.)  Elliptic  elements  have  in  like  manner  been  assigned  to  the 
comet  discovered  by  M.  Brorsen,  on  the  26th  of  February,  1846,  which, 
like  that  last  mentioned,  speedily  after  its  discovery  j.gan  to  show  evident 
symptoms  of  deviation  from  a  parabola.  These  elements,  with  the  names 
of  their  respective  calculators,  are  as  follow.  The  dates  are  for  February 
1846,  Greenwich  time. 


Computed  by 

Brunnow. 

Hind. 

Van  WIIIing:en 
and  De  Ilnan. 

Perihelion  nassaire 

25"-  37794 

116°  28'  34"-0 

102    39    36-  6 

30     55      6-  5 

3-15021 

0-79363 

2042 

25'>-  33109 

116°  28'  17"-8 

102     45    20-  9 

30     49      3-  6 

3-12292 

0-797  <  1 

2016 

25<»-  02227 

116°  23'  62"-9 

103     31    25-  7 

30     30    .SO-  2 

2-87062 

0-77313 

1776 

Loner,  of  Perihelion 

Long,  of  U 

Inclination 

Semiaxis 

£xGentricitv *•■••« 

Period  ^davs)  

COMET  07  DE  VICO. 


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OUTLINES   OP  ASTRONOMY. 


This  comet  is  faint,  and  presents  nothing  rcraarkablo  in  its  appearance. 
Its  chief  interest  arises  from  the  groat  similarity  of  its  parabolic  clemeiit.H 
to  those  of  the  comot  of  1582,  tho  place  of  tho  perihelion  and  node,  aud 
tbo  inclination  of  tho  orbit,  being  almost  identical. 

(688.)  Elliptic  elenients  have  also  been  calculated  by  M.  D' Arrest,  for 
a  comet  discovered  by  M.  Peters,  on  tho  20th  of  Juno,  1840,  whiih  go 
to  assign  it  a  place  among  tho  comets  of  short  period,  viz.  SSO-t^-S,  or 
very  nearly  10  years.  The  cxcentricity  of  tho  orbit  is  0-75672,  its  scnij. 
axis  0-32000,  and  thn  inclination  of  its  piano  to  that  of  tho  ecliptic  31°  2' 
14".     This  comet  passed  its  perihelion  on  tho  1st  of  June,  1840. 

(589.)  By  far  tho  most  remarkable  comet,  however,  which  has  beeu 
seen  during  the  present  century,  is  that  which  appeared  in  tho  spriny  of 
1843,  and  whoso  tail  became  visible  in  the  twilight  of  tho  17th  of  j\Iarch 
in  England  as  a  great  beam  of  nebulous  light,  extending  from  a  poiut 
above  tho  western  horizon,  through  tho  stars  of  Eridanus  and  Lopus, 
under  the  belt  of  Orion.  This  situation  was  low  and  unfavourable ;  and 
it  was  not  till  tho  19th  that  the  head  was  seen,  and  then  only  as  a  fain 
and  ill-defined  nebula,  very  rapidly  fading  on  subsequent  nights.  In 
more  southern  latitudes,  however,  not  only  tho  tail  was  seen,  as  a  mag- 
nificent train  of  light  extending  50"  or  60"  in  length ;  but  tho  head  and 
nucleus  appeared  with  extraordinary  splendour,  exciting  in  every  country 
where  it  was  seen  the  greatest  astonishment  and  admiration.  Indeed,  all 
descriptions  agree  in  representing  it  as  a  stupendous  spectacle,  such  as  in 
superstitious  ages  would  not  fail  to  have  carried  terror  into  every  bosom. 
In  tropical  latitudes  in  the  northern  hemisphere,  tho  tail  appeared  on  the 
3d  of  March,  and  in  Van  Diemen's  Land,  so  early  as  tho  1st,  the  coniut 
having  passed  its  perihelion  on  tho  27th  of  February.  Already  on  the 
3d  the  head  was  so  far  disengaged  from  the  immediate  vicinity  of  the  sun, 
as  to  appear  for  a  short  time  above  the  horizon  after  sunset.  On  this  day 
when  viewed  through  a  40-inch  achromatic  telescope  it  presented  a  plan- 
etary disc,  from  which  rays  emerged  in  tho  direction  of  tho  tail.  The  tail 
was  double,  consisting  of  two  principal  lateral  streamers,  making  a  very 
small  angle  with  each  other,  and  divided  by  a  comparatively  dark  line,  of 
the  estimated  length  of  25",  prolonged,  however,  on  the  north  side  by  a 
divergent  streamer,  making  an  angle  of  5°  or  6"  with  the  general  direc- 
tion of  tho  axis,  and  traceable  as  far  as  65°  from  the  head.  A  similar 
though  fainter  lateral  prolongation  appeared  on  the  south  side.  A  Sne 
drawing  of  it  of  this  date  by  C.  P.  Smyth,  Esq.,  of  the  Royal  Observatory, 
C.  G-.  H.,  represents  it  as  highly  symmetrical,  and  gives  the  idea  of  a 
vivid  cone  of  light,  with  a  dark  axis,  and  nearly  rectilinear  sides,  inclosed 
in  a  fainter  cone,  the  sides  of  wbicii  curve  slightly  outwards.     The  light 


-7^- 


OREAT   COMET   OF   1848. 


81T 


of  the  nucleus  at  this  pori<*^  i»  compwod  to  tLat  of  a  star  of  the  first  or 
second  ninguitudo;  and  on  tl.'  llth,  of  tlie  ijyird;  from  which  lime  it 
degraded  in  light  so  rapidly,  that  on  tho  lUth  it  was  invi.siblo  to  the  naked 
CYC,  the  tail  all  the  while  continuing  brilliantly  visiblo,  though  much 
more  so  at  a  distance  from  the  nucleus,  with  whirli,  indeed,  its  connexion 
was  not  then  obvious  to  the  unassisted  sight  —  a  singular  feature  in  tho 
history  of  this  body.  Tho  tail,  subsequent  to  tho  8d,  was,  generally 
speaking,  a  single  straight  or  slightly  curved  broad  band  of  light,  but  on 
the  llth  it  is  recorded  by  Mr.  Clerihew,  who  observed  it  at  Calcutta,  to 
have  shot  forth  a  lateral  tail  nearly  twice  as  long  as  tho  regular  one,  but 
fainter,  and  making  an  angle  of  about  18°  with  its  direction  on  the 
Boutbern  side.  The  projection  of  this  ray  (which  was  not  seen  cither 
before  or  after  the  day  in  question)  to  so  enormous  a  length,  (nearly 
100°)  in  a  single  day  conveys  an  impression  of  tho  intensity  of  tho  forces 
acting  to  produce  such  a  velocity  of  material  transfer  through  space,  such 
as  no  other  natural  phajnoraenon  is  capable  of  exciting.  It  is  clear  that 
if  we  have  to  deal  here  with  matter,  such  as  we  conceive  it,  viz.  possessing 
vicrtia — at  all,  it  must  be  under  tho  dominion  of  forces  incomparably 
more  energetic  than  gravitation. 

(590.)  There  is  abundant  evidence  of  the  comet  in  question  having 
been  seen  in  full  daylight,  and  in  the  sun's  immediate  vicinity.  It  was 
so  seen  on  the  28th  of  February,  the  day  after  its  perihelion  passage,  by 
every  person  on  board  tho  H.  E.  I.  C.  S.  Owen  Glendower,  then  off  tho 
Cape,  as  a  short,  dag^'^er-like  object,  close  to  the  sun,  a  little  before  sunset. 
On  the  same  day,  at  b"  6"  P.  M.,  and  consequently  in  full  sunshine,  tho 
distance  of  tho  nucleus  from  the  sun  was  actually  measured  with  a  sex- 
tant by  Mr.  Clarke,  of  Portland,  United  States,  the  distance,  centre  from 
centre,  being  then  only  3°  50'  43".  He  describes  it  in  the  following 
terms :  "  The  nucleus,  and  also  every  part  of  the  tail,  were  as  well  defined 
as  the  moon  on  a  clear  day.  The  nucleus  and  tail  bore  the  same  appear- 
ance, and  resembled  a  perfectly  pure  white  cloud,  without  any  variation, 
except  a  slight  change  near  the  head,  just  sufficient  to  distinguish  tho 
nucleus  from  the  tail  at  that  point."  The  denseness  of  the  nucleus  was 
so  considerable,  that  Mr.  Clarke  had  no  doubt  it  might  have  been  visible 
upon  the  sun's  disc,  had  it  passed  between  that  and  the  observer.  Tho 
length  of  the  visible  tail  resulting  from  these  measures  was  59',  or  not  far 
from  double  the  apparent  diameter  of  the  sun ;  and  as  we  shall  presently 
see  that  on  the  day  in  question  the  distance  from  the  earth  of  the  sun  and 
comet  must  have  been  very  nearly  equal,  this  gives  us  about  1700000 
miles  for  the  linear  dimensions  of  this,  the  densest  portion  of  that  ap- 


iii 


i;i' 


'^'■A 

^m 


318 


OUTLINES  OF  ASTRONOMY. 


pendage,  makiag  no  allowance  for  the  foreshortening,  which  at  that  time 
was  very  considerable. 

(591.)  The  elements  of  this  comet  are  among  the  most  remarkable  of 
any  recorded.  They  have  been  calculated  by  several  eminent  astronomers, 
among  whose  results  we  shall  specify  only  those  which  agree  best ;  the 
earlier  attempts  to  compute  its  path  having  been  rendered  uncertain  by 
the  difficulty  attending  exact  observations  of  it  in  the  first  part  of  its 
visible  career.  The  following  are  those  which  seem  entitled  to  most 
confidence : 


Encke. 

Plantamour. 

Knorre. 

Nlcolai. 

Peters. 

Perihel.  pass.,  1843, 
Feb..meantlineat 
Greenwicli     

27"!  45096 

279°  2'  30" 

4  15     5 

35  12  38 

0-00522 

Retrograde. 

27<i-42935 
278°  18'    3" 

0    51     4 
35      8   56 

0-00581 
Retrograde. 

27<i-39G38 
278°  28'  25" 

1    48     3 
35    35  29 

0-00579 
Retrograde. 

27'143023 
278°  36'  33" 

1    37   55 
35    36  29 

0-00568 
Retrograde. 

27'i-41319 
279°  .5!)'    7" 

3    53   17 
35    15  42 

0-0042S 
Retrograde. 

Long,  of  porihel 

Lonir.  of  Q 

Inc1iiia.tion 

Pprihel.  disk 

Motion 

(592.)  What  renders  these  elements  so  remarkable  is  the  sraallness  of 
the  perihelion  distance.  Of  all  comets  which  have  been  recorded  this  has 
made  the  nearest  approach  to  the  sun.  The  sun's  radius  being  the  sine 
of  his  apparent  semi-diameter  (16'  1"  -5)  to  a  radius  equal  to  the  earth's 
mean  distance=l,  is  represented  on  that  scale  by  0-00466,  which  falls 
short  of  0*00534,  the  perihelion  distance  found  by  taking  a  mean  of  all 
the  foregoing  results,  by  only  0-00067,  or  about  one  seventh  of  its  whole 
magnitude.  The  comet,  therefore,  approached  the  luminous  surface  of 
the  sun  within  about  the  seventh  part  of  the  sun's  radius !  It  is  worth 
while  to  consider  what  is  implied  in  such  a  fact.  In  the  first  place,  the 
intensity  both  of  the  light  and  radiant  heat  of  the  sun  at  different  dis- 
tances from  that  luminary  increase  proportionally  to  the  spherical  area  of 
the  portion  of  the  visible  hemisphere  covered  by  the  sun's  disc.  This 
disc,  in  the  case  of  the  earth,  at  its  mean  distance  has  an  angular  dia- 
meter of  32'  8".  At  our  comet  in  perihelio  the  apparent  angular  dia- 
meter of  the  sun  was  no  less  than  121°  32'.  The  ratio  of  the  spherical 
surfaces  thus  occupied  (as  appears  from  spherical  geometry)  is  that  of  the 
squares  of  the  sines  of  the  fourth  parts  of  these  angles  to  each  other,  or 
that  of  1  :  47042.  And  in  this  proportion  are  to  each  other  the  amounts 
of  light  and  heat  thrown  by  the  sun  on  an  equal  area  of  exposed  surface 
on  our  earth  and  at  the  comet  in  equal  instants  of  time.  Let  any  one 
imagine  the  ef-  Jt  of  so  fierce  a  glare  as  that  of  47000  suns  such  as  we 
experience  the  warmth  of,  on  the  materials  of  which  the  earth's  surface 
is  composed.     To  form  some  practical  idea  of  it  we  may  compare  it  with 


GREAT  COMET  OF  1843.  319 

what  is  recorded  of  Parker's  great  lens,  whose  diameter  was  32^  inches 
and  focal  length  six  feet  eight  inches.  The  effect  of  this,  supposing  all 
the  light  and  heat  transmitted,  and  the  focal  concentration  perfect,  (both 
conditions  very  imperfectly  satisfied,)  would  be  to  enlarge  the  sun's 
effective  angular  diameter  to  23°  26',  which,  compared  on  the  same 
principle  with  a  sun  of  32'  in  diameter,  would  give  a  multiplier  of  only 
1915  instead  of  47000.  The  heat  to  which  the  comet  was  subjected 
therefore  surpassed  that  in  the  focus  of  the  lens  in  question,  on  the 
lowest  calculation,  in  the  proportion  of  24^  to  1.  Yet  that  lens  melted 
cornelian,  agate,  and  rock  crystal ! 

(598.)  To  this  extremity  of  heat  however  the  comet  was  exposed  but 
for  a  short  time.  Its  actual  velocity  in  perihelio  was  no  less  than  366 
miles  per  second,  and  the  whole  of  that  segment  of  its  orbit  above  (i.  e. 
north  of)  the  plane  of  the  ecliptic,  and  in  which,  as  will  appear  from  a 
consideration  of  the  elements,  the  perihelion  was  situated,  was  described 
in  little  more  than  two  hours ;  such  being  the  whole  duration  of  the  time 
from  the  ascending  to  the  descending  node,  or  in  which  the  comet  had 
north  latitude.  Arrived  at  the  descending  node,  its  distance  from  the 
sun  would  be  already  doubled,  and  the  radiation  reduced  to  one  fourth 
of  its  maximum  amount.  The  comet  of  1680,  whose  perihelion  distance 
was  0  0062,  and  which  therefore  approached  the  sun's  surface  within  one 
third  part  of  his  radius  (more  than  double  the  distance  of  the  comet  now 
in  question)  was  computed  by  Newton  to  have  been  subjected  to  an 
intensity  of  heat  2000  times  that  of  red-hot  iron, — a  term  of  comparison 
indeed  of  a  very  vague  description,  and  which  modern  thermotics  do  not 
recognize  as  affording  a  legitimate  measure  of  radiant  heat.' 

(594.)  Although  some  of  the  observations  of  this  comet  were  vague 
and  inaccurate,  yet  there  seem  good  grounds  for  believing  that  its  whole 
course  cannot  be  reconciled  with  a  parabolic  orbit,  and  that  it  really 
describes  an  ellipse.  Previous  to  any  calculation,  it  was  remarked  that 
in  the  year  1668  the  tail  of  an  immense  comet  was  seen  in  Lisbon,  at 
Bologna,  in  Brazil,  and  elsewhere,  occupying  nearly  the  same  situation 
among  the  stars,  and  at  the  same  season  of  the  year,  viz.  on  the  5th  of 
March  and  the  following  days.     Its  brightness  was  such  that  its  reflected 

'  A  transit  of  this  comet  over  the  sun's  disc  must  probably  have  taken  place  shortly 
after  its  passage  through  its  decending  node.  It  is  greatly  to  be  regretted  that  so 
interesting  a  pheenomenon  should  have  passed  unobserved.  Whether  it  be  possible 
that  some  offset  of  its  tail,  darted  off  so  late  as  the  7th  of  March,  when  the  comet 
was  already  far  south  of  the  ecliptic,  should  have  crossed  that  plane  and  been  seen 
near  the  Pleiades,  may  be  doubted.  Certain  it  is,  that  on  the  evening  of  that  day,  a 
decidedly  cometic  ray  was  seen  in  the  immediate  neighbourhood  of  those  stars  by  Mr. 
Nasmyth.  (Ast  Soc.  Notices,  vol.  v.  p.  270.) 


^'mi 


320 


OUTLINES   OF  ASTRONOMY. 


trace  was  easily  distinguished  on  the  sea.  The  head,  when  it  at  length 
came  in  sight,  was  comparatively  faint  and  scarce  discernible.  No  precise 
observations  were  made  of  this  comet,  but  the  singular  coinciderce  of 
situation,  season  of  the  year,  and  physical  resemblance,  excited  a  strong 
suspicion  of  the  identity  of  the  two  bodies,  implying  a  period  of  175 
years  loithin  a  day  or  two  more  or  less.  This  suspicion  has  been  con- 
verted almost  into  a  certainty  by  a  careful  examination  of  what  is  recorded 
of  the  older  comet.  Locating  on  a  celestial  chart  the  situation  of  the 
head,  concluded  from  the  direction  and  appearance  of  the  tail,  when  only 
that  was  seen,  and  its  visible  place,  when  mentioned,  according  to  the 
descriptions  given,  it  has  been  found  practicable  to  derive  a  rough  orbit 
from  the  course  thus  laid  down :  and  this  agrees  in  all  its  features  so  well 
with  that  of  the  modem  comet  as  nearly  to  remove  all  doubt  on  the 
subject.  Comets,  moreover,  are  recorded  to  have  been  seen  in  a.  d.  268, 
442-3,  791,  968,  1143,  1317,  1494,  which  may  have  been  returns  of 
this,  since  the  period  above-mentioned  would  bring  round  its  appearance 
to  the  years  268,  443,  618,  793,  968,  1143,  1318,  and  1493,  and  a 
certain  latitude  must  always  be  allowed  for  unknown  perturbations. 

(595.)  But  this  is  not  the  only  comet  on  record  whose  identity  with 
the  comet  of  '43  has  been  maintained.  In  1689  a  comet  bearing  a  con* 
siderable  resem]blance  to  it  was  observed  from  the  8th  to  the  23d  of  De- 
cember, and  from  the  few  and  rudely  observed  places  recorded,  its  elements 
had  been  calculated  by  Pingr6,  one  of  the  most  diligent  inquirers  into  this 
part  of  astronomy.'  From  these  it  appears"  that  the  perihelion  distance 
of  that  comet  was  very  remarkably  small,  and  a  sufficient  though  indeed 
rough  coincidence  in  the  places  of  the  perihelion  and  node  tended  to 
corroborate  the  suspicion.  But  the  inclination  (69°)  assigned  to  it  by 
Pingre  appeared  conclusive  against  it.  On  recomputing  the  elements, 
however,  from  his  data.  Professor  Pierce  has  assigned  to  that  comet  an 
inclination  widely  differing  from  Pingre* s,  viz.  30°  4'",  and  quite  within 
reasonable  limits  of  resemblance.  But  how  doe?  this  agree  with  the 
longer  period  of  175  years  before  assigned  ?    To  reconcile  this  we  must 

'  Author  of  the  "  Cometographie,"  a  work  indispensable  to  all  who  would  study 
this  interesting  department  of  the  science. 

•United  States  Gazette,  May  29,  1843.  Considering  that  all  the  observations  lie 
near  the  descending  node  of  the  orbit,  the  proximity  of  the  comet  at  that  time  to  the 
sun,  and  the  loose  nature  of  the  recorded  observations,  no  doubt  almost  any  given  in- 
clination might  be  deduced  from  them.  The  true  test  in  such  cases  is  not  to  ascend 
from  the  old  incorrect  data  to  elements,  but  to  descend  from  known  and  certain  ele- 
ments to  the  older  data,  and  ascertain  whether  the  recorded  phenomena  can  be  repre- 
iiented  by  them  (perturbations  included)  within  fair  limits  of  interpretation.  Such  is 
the  course  pursued  by  Clausen. 


GREAT   COMET   OF  1843. 


821 


suppose  that  these  175  years  comprise  at  least  eight  returns  of  the  comet, 
and  that  in  effect  a  mean  period  of  21y875  must  be  allowed  for  its  return. 
Now  it  is  worth  remarking  that  this  period  calculated  backwards  from 
1843-156  will  bring  us  upon  a  series  of  years  remarkable  for  the  appear- 
ance of  great  comets,  many  of  which,  as  well  as  the  imperfect  descriptions 
we  have  of  their  appearance  and  situation  in  the  heavens,  offer  at  least  no 
obvious  contradiction  to  the  supposition  of  their  identity  with  this.  Be- 
sides those  already  mentioned  as  indicated  by  the  period  of  175  years,  we 
may  specify  as  probable  or  possible  intermediate  returns,  those  of  the 
comets  of  1733?',  1689  above-mentioned,  1559?,  1537^  1515»,  1471, 
1426, 1405-6,  1383,  1361, 1340^  1296, 1274, 1230^  1208, 1098, 1056, 
1034, 1012«,  990  ?^  925?,  858??,  G84«,  552,  530»,  421,  245  or  247'°, 
180",  158.  Should  this  view  of  the  subject  be  the  true  one,  we  may  ex- 
pect its  return  about  the  end  of  1864  or  beginning  of  1865,  in  which 
event  it  will  be  observable  in  the  Southern  Hemisphere  both  before  and 
after  its  perihelion  passage". 

(596.)  M.  Clausen,  from  the  assemblage  of  all  the  observations  of  this 
comet  known  to  him,  has  calculated  elliptic  elements  which  give  the  extra- 
ordinarily short  period  of  6 -38  years.  And  in  effect  it  has  been  suggested 
that  a  still  further  subdivision  of  the  period  of  21-875  into  three  of 
7-292  years,  would  reconcile  this  with  other  remarkable  comets.  This 
seems  going  too  far ;  but  at  all  events  the  possibility  of  representing  its 
motions  by  so  short  an  ellipse  will  easily  reconcile  us  to  the  admission  of 
a  period  of  21  years.  That  it  should  only  be  visible  in  certain  apparitions, 
and  not  in  others,  is  sufficiently  explained  by  the  situation  of  its  orbit. 

(597.)  We  have  been  somewhat  diffuse  on  the  subject  of  this  comet. 


This  great  southern  comet  of  May  17th  seems  too  early  in 


In  January  1537,  a  comet  was  seen  in  Pisces. 

A  comet  predicted  the  death  of  Ferdinand  the  Catholic. 


He  died 


'  P.  Passage  1733-781 
the  year. 

'  F.  P.  1536-906. 

»P.  P.  1515031. 
Jan.  23,  1515. 

*  P.  P.  1340  031.    Evidently  a  southern  comet,  and  a  very  probable  appearance. 

'  P.  P.  1230-656,  was  perhaps  a  return  of  Halley's. 

"  P.  P.  1011-906.  In  1012,  a  very  great  comet  in  the  southern  part  of  the  heavens. 
"  f'oii  eclat  blessait  les  yeux."  (Pingre  Cometographie,  from  whom  all  these  recorded 
appearances  are  taken.) 

'  P.  P.  990-031.     "  Comete  fort  epouvantable,"  some  year  between  989  and  998. 

'  P.  P.  683-781.   In  684,  appeared  two  or  three  comets.    Dates  begin  to  be  obscure. 

"  Two  distinct  comets  (one  probably  the  comet  of  Caesar  and  1680)  appeared  in  530 
and  531,  the  former  observed  in  China,  the  latter  in  Europe. 

"•  P.  P.  246-281 ;  both  southern  comets  of  the  Chinese  annals.  The  year  of  one  or 
other  may  be  wrong. 

"  P.  P.  180-656.  Nov.  6,  a.d.  180.    A  southern  comet  of  the  Chinese  annals. 

"  Clausen,  Astron.  Nachr.  No.  485. 

21 


«'   ^M 


''"'  IE 


822 


OUTLINES   OF  ASTRONOMY. 


for  the  sake  of  showiug  the  degree  and  kind  of  interest  which  attaches  to 
cometio  astronomy  in  the  present  state  of  the  science.  In  fact  there  is  no 
branch  of  astronomy  more  replete  with  interest,  and  we  may  add  more 
eagerly  pursued  at  present,  inasmuch  as  the  hold  which  exact  calculaiion 
gives  us  on  it  may  be  regarded  as  completely  established ;  so  that  whatever 
may  be  concluded  as  to  the  motions  of  any  comet  which  shall  hencefor- 
ward come  to  be  observed,  will  be  concluded  on  sure  grounds  and  with 
numerical  precision ;  while  the  improvements  which  have  been  introduced 
into  the  calculation  of  cometary  perturbation,  and  the  daily  increasing 
familiarity  of  numerous  astronomers  with  computations  of  this  nature, 
enable  us  to  trace  their  past  and  future  history  with  a  certainty,  which  at 
the  commencement  of  the  present  century  could  hardly  have  been  looked 
upon  as  attainable.  Every  comet  newly  discovered  is  at  opce  subjected  to 
the  ordeal  of  a  most  rigorous  inquiry.  Its  elements,  roughly  calculated 
within  a  few  days  of  its  appearance,  are  gradually  approximated  to  as  ob- 
servations accumulate,  by  a  multitud*^  of  ardent  and  expert  computists. 
On  the  least  indication  of  a  deviation  from  a  parabolic  orbit,  its  ellipt'c 
elements  become  a  subject  of  universal  and  lively  interest  and  discussion, 
Old  records  are  ransacked,  with  all  the  advantage  of  improved  data  and 
methods,  so  as  to  rescue  from  oblivion  the  orbits  of  ancient  comets  which 
present  any  oimjlarity  to  that  of  the  new  visitor.  The  disturbances  under- 
gone in  the  interval  by  the  action  of  the  planets  are  investigated,  and  the 
past,  thus  brought  into  unbroken  connexion  with  the  present,  is  made  to 
afford  substantial  ground  for  prediction  of  the  future.  A  great  impulse, 
meanwhile,  has  been  given  of  late  years  to  the  discovery  of  comets,  by 
the  establishment  in  1840',  by  his  late  Majesty  the  King  of  Denmark, 
of  a  prize  medal,  to  be  awarded  for  every  such  discovery,  to  the  first  ob- 
server, (the  influence  of  which  may  be  most  unequivocally  traced  in  the 
great  number  of  these  bodies  which  every  successive  year  sees  added  to 
our  list,)  and  by  the  circulation  of  notices,  by  special  letter*,  of  every  such 
discovery  (accompanied,  when  possible,  by  an  ephemeris),  to  all  observers 
who  have  shewn  that  they  take  an  interest  in  the  inquiry,  so  as  to  ensure 
the  full  and  complete  observation  of  the  new  comet,  so  long  as  it  remains 
within  the  reach  of  our  telescopes. 

(598.)  It  is  by  no  means  merely  as  a  subject  of  antiquarian  interest,  or 
on  account  of  ihe  brilliant  spectacle  which  comets  occasionally  afford,  that 
ftstronomers  attach  a  high  degree  of  importance  to  all  that  regards  them. 
Apart  even  from  the  singularity  and  mystery  which  appertains  to  their 


*  See  the  announcement  of  this  institution  in  Astron.  Nachr.  No.  400. 
'  By  Prof.  Schumacher,  Director  of  the  Royal  Observatory  of  Ahona. 


INTEREST  ATTACHED  TO   COMETART  ASTRONOMY. 


823 


physical  constitution,  they  have  become,  through  the  medium  of  exact 
calculation,  unexpected  instruments  of  inquiry  into  points  connected  with 
the  planetary  system  itself,  of  no  small  importance.  We  have  seen  that 
the  movements  of  the  comet  of  Encke,  thus  minutely  and  perseveringly 
traced  by  the  eminent  astronomer  whose  name  is  used  to  distinguish  it, 
has  afforded  ground  for  believing  in  the  presence  of  a  resisting  medium 
filling  the  whole  of  our  system.  Similar  inquiries,  prosecuted  in  the 
cases  of  other  periodical  comets,  will  extend,  conf  rm,  or  modify  our  con- 
clusions on  this  head.  The  perturbations,  too,  which  comets  experience 
in  passing  near  any  of  the  planets,  may  afford,  and  have  afforded,  infor- 
mation as  to  the  magnitude  of  the  disturbing  masses,  which  could  not  well 
be  otherwise  obtained.  Thus  the  approach  of  this  comet  to  the  planet 
Mercury  in  1838  afforded  an  estimation  of  the  mass  of  that  planet,  the 
more  precious,  by  reason  of  the  great  uncertainty  under  which  all  previous 
determinations  of  that  element  laboured.  Its  approach  to  the  same  planet 
in  the  present  year  (1848)  will  be  still  nearer.  On  the  22d  of  November 
their  mutual  distance  will  be  only  fifteen  times  the  moon's  distance  from 
the  earth. 

(599.)  It  is,  however,  in  a  physical  point  of  view  that  these  bodies  offer 
the  greatest  stimulus  to  our  curiosity.  There  is,  beyond  question,  some 
profound  secret  and  mystery  of  nature  concerned  in  the  phaenomenon  of 
their  tails.  Perhaps  it  is  not  too  much  to  hope  that  future  observation, 
borrowing  every  aid  from  rational  speculation,  grounded  on  the  progress  of 
physical  science  generally,  (especially  those  branches  of  it  which  relate  to 
the  sethereal  or  imponderable  elements),  may  ere  long  enable  us  to  pene- 
trate this  mystery,  and  to  declare  whether  it  is  really  matter,  in  the  ordi- 
nary acceptation  of  the  term,  which  is  projected  from  their  heads  with 
such  extravagant  velocity,  and  if  not  impelled,  at  least  directed  in  its 
course  by  a  reference  to  the  sun,  as  its  point  of  avoidance.  In  no  respect 
is  the  question  as  to  the  materiality  of  the  tail  more  forcibly  pressed  on  us 
for  consideration,  than  in  that  of  the  enormoas  sweep  which  it  makes 
round  the  sun  in  perihelio,  in  the  ~ianner  of  a  i;traight  and  rigid  rod,  in 
defiance  of  the  law  of  gravitation,  nay,  even  of  the  received  laws  of  mo- 
tion, extending  (as  we  have  seen  in  the  comets  of  1680  and  1843)  from 
near  the  sun's  surface  to  the  earth's  orbit,  yet  whirled  round  unbroken  j 
in  the  latter  case  through  an  angle  of  180°  in  little  more  than  two  hours. 
It  seems  utterly  incredible  that  in  such  a  case  it  is  one  and  the  same 
material  object  which  is  thus  brandished.  If  there  could  be  c6nceived 
such  a  thing  as  a  negative  shadow,  a  momentary  impression  made  upon 
the  luminiferous  aether  behind  the  comet,  this  would  represent  in  some 
degree  the  conception  such  a  phfenomenon  irresistibly  calls  up.     But  thitf 


7! 


^^. 


■ '     f  '  »- 'fin 


'f?i 


324 


OUTLINES  OF  ASTRONOMY. 


is  not  all.  Even  such  an  extraordinary  excitement  of  the  SDther,  conceive 
it  as  we  will,  will  afford  no  account  of  the  projection  of  lateral  streamers ; 
of  the  effusion  of  light  from  the  nucleus  of  a  comet  towards  the  sun ;  and 
its  subsequent  rejection ;  of  the  irregular  and  capricious  mode  in  which 
that  effusion  has  been  seen  to  take  place ;  none  of  the  clear  indications  of 
alternate  evaporation  and  condensation  going  on  in  the  immense  regions  of 
space  occupied  by  the  tail  and  coma,  —  none,  in  short,  of  innumerable 
other  facts  which  link  themselves  with  almost  equally  irresistible  cogency 
to  our  ordinary  notions  of  matter  and  force.  '-  '     ' 

(600.)  The  great  number  of  comets  which  appear  to  move  in  parabolic 
orbits,  or  orbits  at  least  undistinguishable  from  parabolas  during  their  de- 
scription of  that  comparatively  small  part  within  the  range  of  their  visi- 
bility to  us,  has  given  rise  to  an  impression  that  the}  are  bodies  extraneous 
to  our  system,  wandering  through  space,  and  merely  yielding  a  local  and 
temporary  obedience  to  its  laws  during  their  sojourn.  What  truth  there 
may  be  in  this  view,  we  may  never  have  satisfactory  grounds  for  deciding. 
On  such  an  hypothesis,  our  elliptic  comets  owe  their  permanent  deuizoi- 
ship  within  the  sphere  of  the  sun's  predominant  attraction  to  the  action 
of  one  or  other  of  the  planets  near  which  they  may  have  passed,  in  such 
a  manner  as  to  diminish  their  velocity,  and  render  it  compatible  with 
elliptic  motion.'  A  similar  cause  acting  the  other  way,  might  with  equal 
probability,  give  rise  to  a  hyperbolic  motion.  But  whereas  in  the  former 
case,  the  comet  would  remain  in  the  system,  and  might  make  an  indefinite 
number  of  revolutions,  in  the  latter  it  would  return  no  more.  This  may 
possibly  be  the  cause  of  the  exceedingly  rare  occurrence  of  a  hyperbolic 
comet  as  compared  with  elliptic  ones. 

(601.)  All  the  planets  without  exception,  and  almost  all  the  satellites, 
circulate  in  one  direction.  Retrograde  comets,  however,  are  of  very  com- 
mon occurrence,  which  certainly  would  go  to  assign  them  an  exterior  or 
at  least  an  independent  origin.  Laplace,  from  a  consideration  of  all  the 
cometary  orbits  known  in  the  earlier  part  of  the  present  century,  con- 
cluded that  the  mean  or  average  situation  of  the  planes  of  all  the  cometary 
orbits,  with  respect  to  the  ecliptic,  was  so  nearly  that  of  perpendicularity, 
as  to  afford  no  presumption  of  any  cause  Massing  their  directions  in  this 
respect.  Yet  we  think  it  worth  noticing,  that  among  the  comets  which 
are  as  yet  known  to  describe  elliptic  orbits,  not  one  whose  inclination  is 
under  17°  is  retrograde;  and  that  out  of  thirty-six  comets  which  have 
had  elliptic  elements  a&.^igned  to  them,  whether  of  gieat  or  small  excen- 
tricities,  and  without  any  limit  of  inclination,  only  five  are  retrograded, 

'  The  velocity  in  an  ellipse  is  always  less  than  in  a  parabola,  at  equal  distances  from 
the  sun :  in  an  hyperbola  always  greater.  I  \ 


■I  •■ 


■■'Q 


GENERAL  REMARKS. 


825 


and  of  these  only  two,  viz.  Halley's  and  the  grea^  con-et  of  1843,  can  be 
regarded  as  satisfactorily  made  out.  Finally,  of  the  125  comets  whose 
elements  arc  given  in  the  collection  of  Schumacher  and  Olbers,  up  to 
1823,  the  number  of  retrograde  comets  under  10°  of  inclination  is  only 
2  out  of  9,  and  under  20°,  7  out  of  23.  A  plane  of  motion,  therefore, 
nearly  coincident  with  the  ecliptic,  and  a  periodical  return,  are  circum- 
stances eminently  favourable  to  direct  revolution  in  the  cometary  as  they 
are  decisive  among  the  planetary  orbits.  [Here  also  we  may  notice  a  very 
curious  remark  of  Mr.  Hind,  (Ast.  Nachr.  No.  724,)  respecting  periodic 
comets,  viz.,  that,  so  far  as  at  present  known,  they  divide  themselves  for 
the  most  part  into  two  families,  —  the  one  having  periods  of  about  75 
years,  corresponding  to  a  mean  distance  about  that  of  Uranus ;  the  other 
corresponding  more  nearly  with  those  of  the  asteroids,  and  with  a  mean 
distance  between  those  small  planets  and  Jupiter.  The  former  group 
consists  of  four  members,  Halley's  comet  revolving  in  76  years,  one  dis- 
covered by  Olbers  in  74,  De  Vico's  4th  comet  in  73,  and  Brorsen's  3d  in 
75,  respectively.  Examples  of  the  latter  group  are  to  be  seen  in  the 
table,  p.  652,  at  the  rt'  ^  of  this  volume.  It  may  be  added,  also,  ♦hat 
one  or  two  of  the  asteroids  are  described  as  having  a  faint  nebulous  enve- 
lope about  them,  indicating  somewhat  of  a  cometic  nature.] 


1  >  ^  yi 


l;:f 


i  M^ 


826 


OUTLINES   OF  ASTRONOMY. 


PART  II. 

OF  THE   LUNAR  AND   PLANETARY  PERTURBATIONS. 
"  Magnus  ab  integro  saBclcruin  nascitur  ordo." — Viro.  Pollio. 

CHAPTER  XII. 

SUBJECT  PROPOUUDED. — PROBLEM  OP  THREE  BODIES. — SUPERPOSITION 
OF  SMALL  MOTIONS.  —  ESTIMATION  OP  THE  DISTURBING  FORCB.— 
ITS  GEOMETRICAL  REPRESENTATION.  —  NUMERICAL  ESTIMATION  IN 
PARTICULAR  CA8FS.  —  RESOLUTION  INTO  RECTANGULAR  COMPO- 
NENTS. —  RADIAL,  TRANSVERSAL,  AND  ORTHOGONAL  DISTURBING 
FORCES.  —  NORMAL  AND  TANGENTIAL.  —  THEIR  CHARACTERISTIC 
EFFECTS. — EFFECTS  OP  THE  ORTHOGONAL  FORCE. — MOTION  OF  THE 
NODES. —  CONDITIONS  OP  THEIR  ADVANCE  AND  RECESS. —  CASES  OF 
AN  EXTERIOR  PLANET  DISTURBED  BY  AN  INTERIOR. — THE  REVERSE 
CASE. —  IN  E""ERY  CASE  THE  NODE  OP  THE  DISTURBED  ORBIT  RE- 
CEDES ON  THE  PLANE  OP  THE  DISTURBING  ON  AN  AVERAGE.— 
COMBINED  EFFECT  OP  MANY  SUCH  DISTURBANCES. — MOTION  OF  THE 
moon's    nodes.  —  CHANGE  OP  INCLINATION.  —  CONDITIONS  OP  ITS 

'  INCREASE  AND  DIMINUTION. — AVERAGE  EFFECT  IN  A  WHOLE  RE- 
VOLUTION.—  COMPENSATION  IN  A  COMPLETE  REVOLUTION   OF  THE 

NODES. — Lagrange's  theorem  of  the  stability  op  the  incli- 
nations OP  THE  PLANETARY  ORBITS. —  CHANGE  OP  OBLIQUITY  OF 
the  ECLIPTIC. — PRECESSION  OP  THE  EQUINOXES  EXPLAINED.— 
NUTATION.  —  PRINCIPLE  OP   FORCED   VIBRATIONS. 


(602.)  In  the  progress  of  this  work,  we  hk:ve  more  than  once  called  the 
reader's  attention  to  the  existence  of  inequalities  in  the  lunar  and  plane- 
tary motions  not  included  in  the  expression  of  Kepler's  Sws,  but  in  some 
sort  supplementary  to  them,  and  of  an  order  so  far  subordinate  to  those 
leading  features  of  the  celestial  movements,  as  to  require,  for  their  detec- 
Kon,  nicer  observations,  and  longer  continued  "lomparison  between  facts 


PERTURBATIONS. 


827 


and  theories,  than  suffice  for  the  establishment  and  verification  of  the 
elliptic  theory.  These  inequalities  arc  known,  in  physical  astronomy,  by 
the  name  of  perturbations.  They  aris'^  in  the  case  of  the  primary 
planets,  from  the  mutual  gravitations  of  (bese  planets  towards  each  other, 
which  derange  their  elliptic  motions  round  the  sun;  and  in  that  of  the 
secondaries,  partly  from  the  n  itual  gravitation  of  the  secondaries  of  the 
same  system  similarly  deranging  their  elliptic  motions  round  their  common 
primary,  and  partly  from  the  unequal  attraction  of  the  sun  and  planets  on 
tbcni  and  on  their  primary.  These  perturbations,  although  small,  and,  in 
most  instances,  insensible  in  short  intervals  of  time,  yet,  when  accumu- 
lated, as  some  of  them  may  become,  in  the  lapse  of  ages,  alter  very 
greatly  the  original  elliptic  relations,  so  as  to  render  the  same  elements  of 
the  planetary  orbits,  which  at  one  ej-och  represented  perfectly  well  their 
movenicnti:',  inadequate  and  unsatisfactory  after  long  intervals  of  time. 

(603.)  When  Newton  first  reasoned  his  way  from  the  broad  features 
of  the  celestial  motions,  up  to  the  law  of  universal  gravitation,  as  affect- 
ing all  matter,  and  rendering  every  particle  in  the  universe  subject  to  the 
influence  of  every  other,  he  was  not  unaware  of  the  modifications  which 
this  generalization  would  induce  upon  the  results  of  a  more  partial  and 
limited  application  of  the  same  law  to  the  revolutions  of  the  planets  about 
the  sun,  and  the  satellites  about  their  primaries,  as  their  onl^  centres  of 
attraction.  So  far  from  it,  his  extraordinary  sagacity  enabled  him  to  per- 
ceive very  distinctly  how  several  of  the  most  important  of  the  lunar 
inequalities  take  their  origin,  in  this  more  general  way  of  conceiving  the 
agency  of  the  attractive  power,  especially  the  retrograde  motion  of  the 
nodes,  and  the  direct  revolution  of  the  apsides  of  her  orbit.  And  if  he 
did  not  extend  his  investigations  to  the  mutual  perturbations  of  the  planets, 
it  was  not  for  want  of  perceiving  that  such  perturbations  must  exist,  and 
might  go  the  length  of  producing  great  derangements  from  the  actual  state 
of  the  system,  but  was  owing  to  the  then  undeveloped  state  o^  the  prac- 
tical part  of  astronomy,  which  had  not  yet  attained  the  precision  requisite 
to  make  such  an  attempt  inviting,  or  indeed  feasible.  What  Newton  left 
undone,  however,  his  successors  have  accomplished ;  and,  at  this  day,  it 
is  hardly  too  much  to  assert  that  there  is  not  a  single  perturbation,  great 
or  small,  which  observation  has  become  precise  enough  clearly  to  detect 
and  place  in  evidence,  which  has  not  been  traced  up  to  its  origin  in  the 
mutual  gravitation  of  the  parts  of  our  system,  and  minutely  accounted 
for,  in  its  numerical  amount  and  value,  by  strict  calculation  on  Newton's 
principles. 

(604.)  Calculations  of  this  nature  require  a  very  high  analysis  for  theii 
successful  performance,  such  as  is  far  beyond  the  scope  and  object  of  this 


i 


m 


4 


I 


"^li 


1 1 


H4 


m^ 


828 


OUTLINES   OF   ASTRONOMY. 


work  to  attempt  exhibiting.  The  reader  who  would  master  then  must 
jToparo  himself  for  the  undertaking  by  an  extensive  course  of  prepara- 
tory study,  and  must  ascend  by  steps  which  wo  must  not  hero  oven  digress 
to  point  out.  It  will  be  our  object,  in  this  chapter,  however,  to  give  souio 
general  insight  into  the  nature  and  manner  of  operation  of  the  acting  forces, 
and  to  point  out  what  are  the  circumstances  which,  in  some  cases,  give  tliciu 
a  high  degree  of  efficiency — a  sort  of  purchase  on  the  balauco  of  tlic  sys- 
tem ;  while,  in  others,  with  no  less  amount  of  intensity,  their  cftoctive 
agtMiey  in  producing  extensive  and  lasting  changes  is  compensated  or  ren- 
dered abortive  J  as  well  as  to  explain  the  nature  of  those  admirable  results 
respecting  the  stability  of  our  system,  to  which  the  researches  of  geome- 
ters have  conducted  them ;  and  which,  under  tho  form  of  matliciuatical 
theorems  of  great  simplicity  and  elegance,  involve  the  history  of  the  past 
and  future  state  of  the  planetary  orbits  during  ages,  of  which,  contem- 
plating the  subject  in  this  point  of  view,  we  neither  perceive  the  begin- 
ning nor  the  end. 

(605.)  Were  there  no  other  bodies  in  the  universe  but  the  sun  and  one 
planet,  the  latter  would  describe  an  exact  ellipse  about  the  former  (or  both 
round  their  common  centre  of  gravity),  and  continue  to  perform  its  revo- 
lutions in  one  and  the  same  orbit  for  ever ;  but  the  moment  we  add  to 
our  combination  a  third  body,  tho  attraction  of  this  will  draw  both  the 
former  bodies  out  of  their  mutual  orbits,  and,  by  acting  on  them  un- 
equally, will  disturb  their  relation  to  each  other,  and  put  an  cud  to  the 
rigorous  and  mathematical  exactness  of  their  elliptic  motions,  not  only 
about  a  fixed  point  in  space,  but  about  one  another.  From  this  way  of 
propounding  the  subject,  we  see  that  it  is  not  the  whole  attraction  of  the 
newly-introduced  body  which  produces  perturbation,  but  the  difference  of 
its  attractions  on  the  two  originally  present. 

(606.)  Compared  to  the  sun,  all  the  planets  are  of  extreme  minuteness; 
the  mass  of  Jupiter,  the  greatest  of  them  all,  being  not  more  than  about 
one  1100th  part  that  of  the  sun.  Their  attractions  on  each  other,  there- 
fore, are  all  very  feeble,  compared  with  the  presiding  central  power,  and 
the  eflFects  of  their  disturbing  forces  are  proportionally  minute.  In  the 
case  of  the  secondaries,  the  chief  agent  by  which  their  motions  are 
deranged  is  the  sun  itself,  whose  mass  is  indeed  great,  but  whose  disturb- 
ing influence  is  immensely  diminished  by  their  near  proximity  to  their 
primaries,  compared  to  their  distances  from  the  sun,  which  renders  the 
difference  of  attractions  on  both  extremely  small,  compared  to  the  whole 
amount.  In  this  case  the  greatest  part  of  the  sun's  attraction,  viz.  that 
which  is  common  to  both,  is  exerted  to  retain  both  primary  and  secondary 
m  their  common  orbit  about  itself,  and  prevent  their  parting  company. 


SUPERPOSITION   OF   SMALL   MOTIONS. 


829 


Only  the  small  overplus  of  force  on  one  as  compared  with  the  other  acta 
08  a  di  rbing  power.  The  mean  value  of  this  ovcrpluH,  in  the  caso  of 
the  moon  disturbed  by  the  sun,  is  calculated  by  Newton  to  amount  to 
no  hij^hor  a  fraction  than  g^e'ooo  of  gravity  at  the  earth's  surface,  or  j^^j 
of  the  principal  force  which  retains  the  moon  in  its  orbit. 

(007.)  From  this  extreme  minuteness  of  the  intensities  of  tho  disturb- 
ing, ciinipared  to  tho  principal  forces,  and  tho  consequent  smallness  of 
their  monientari/  effects,  it  happens  that  wo  can  estimate  each  of  these 
effects  separately,  as  if  tho  others  did  not  trko  place,  without  fear  of 
inducing  error  in  our  conclusions  beyond  the  limits  necessarily  incident 
to  a  llrst  approximation.  It  is  a  principle  in  mechanics,  immediately 
flowing  from  tho  primary  relations  between  forces  and  the  motions  they 
produce,  that  when  a  number  of  very  minuto  forces  act  at  once  on  a 
system,  their  joint  effect  is  tho  sum  or  aggregate  of  their  separate  effects, 
at  least  within  such  limits,  that  the  original  relation  of  the  parts  of  tho 
system  shall  not  have  been  materially  changed  by  their  action.  Such 
effects  supervening  on  tho  greater  movements  due  to  tho  action  of  the 
primary  forces  may  bo  compared  to  the  small  ripplings  caused  by  a 
thousand  varying  breezes  on  the  broad  and  regular  swell  of  a  deep  and 
rolling  ocean,  which  run  on  as  if  the  surface  were  a  plane,  and  cross  in  all 
directions  without  interfering,  each  as  if  the  other  had  no  existence.  It 
is  only  when  their  effects  become  accumulated  in  lapse  of  time,  so  as  to 
alter  the  primary  relations  or  data  of  the  system,  that  it  becomes  neces- 
sary to  have  especial  regard  to  the  changes  correspondingly  introduced 
into  the  estimation  of  their  momentary  efficiency,  by  which  the  rate  of  the 
subsequent  changes  is  affected,  and  periods  or  cycles  of  immense  length 
take  their  origin.  From  this  consideration  arise  some  of  tho  most  curious 
theories  of  physical  astronomy. 

(608.)  Hence  it  is  evident,  that  in  estimating  the  disturbing  influence 
of  several  bodies  forming  a  system,  in  which  one  has  a  remarkable  pre- 
ponderance over  all  the  rest,  we  need  not  embarrass  ourselves  with  combi- 
nations of  the  disturbing  powers  one  among  another,  unless  where 
immensely  long  periods  are  concerned ;  such  as  consist  of  many  hundreds 
of  revolutions  of  the  bodies  in  question  about  their  common  centre.  So 
that,  in  effect,  so  far  as  we  propose  to  go  into  its  consideration,  the 
problem  of  the  investigation  of  the  perturbations  of  a  system,  however 
numerous,  constituted  as  ours  is,  reduces  itself  to  that  of  a  system  of  three 
bodies:  a  predominant  central  body^  a  disturbing,  and  a  disturbed;  the 
two  latter  of  which  may  exchange  denominations,  according  as  the  motions 
of  the  one  or  the  other  are  the  subject  of  inquiry. 

(609.)  Both  the  intensity  and  direction  of  the  disturbing  force  are 


830 


OUTLINES   OF   ASTRONOMY. 


continually  varying,  according  to  the  relative  situation  of  the  disturbing 
and  disturbed  body  with  respect  to  the  sun.  If  the  attraction  of  tiiu  dit)- 
turbiug  body  M,  on  the  central  body  S,  and  the  disturbed  body  1\  (hj 
which  designations,  for  brevity,  we  shall  hereafter  indicate  them,)  were 
equal,  and  acted  in  parallel  linos,  whatever  might  otherwise  be  its  law  of 
variation,  there  would  bo  no  deviation  caused  in  the  elliptic  motion  of  P 
about  S,  or  of  each  about  the  other.  The  case  would  be  strictly  that  of 
art.  454 ;  the  attraction  of  M,  so  circumstanced,  being  at  every  uiouicnt 
exactly  analogous  in  its  effects  to  terrestrial  gravity,  which  acts  in  parallel 
lines,  and  is  equally  intense  on  all  bodies,  great  and  small.  But  this  is 
not  the  case  of  nature.  Whatever  is  stated  in  the  subsequent  artiulo  to 
that  last  cited,  of  the  disturbing  effect  of  the  sun  and  moon,  is,  mutatis 
mutandis,  applicable  to  every  case  of  perturbation ;  and  it  must  bo  now 
our  business  to  enter,  somewhat  more  in  detail,  into  the  general  heads  of 
the  subject  there  merely  hinted  at. 

(610.)  To  obtain  clear  ideas  of  the  manner  in  which  the  di.sturbing 
force  produces  its  various  effects,  we  must  ascertain  at  any  given  moiaeut, 
and  in  any  relative  situations  of  the  three  bodies,  its  direction  and  inten- 
sity as  compared  with  the  gravitation  of  P  towards  S,  in  virtue  of  which 
latter  force  alone  P  would  describe  an  ellipse  about  S  regarded  as  fixed, 
or  rather  P  and  S  about  their  common  centre  of  gravity  in  virtue  of  their 
mutual  gravitation  to  each  other.  In  the  treatment  of  the  problem  of 
thrje  bodies,  it  is  convenient,  and  tends  to  clearness  of  apprehension,  to 
regard  one  of  them  as  fixed,  and  refer  the  motions  of  the  others  to  it  as 
to  a  relative  centre.  In  the  case  of  two  planets  disturbing  each  other's 
motions,  the  sun  is  naturally  chosen  as  this  fixed  centre ;  but  in  ^hat  of 
satellites  disturbing  each  other,  or  disturbed  by  the  sun,  the  centre  of 
their  primary  is  taken  as  their  point  ot  reference,  and  the  sun  itself  is 
regarded  in  the  light  of  a  very  distant  and  massive  satellite  revolving 
about  the  primary  in  a  relative  orbit,  equal  and  similar  to  that  which  the 
primary  describes  absolutely/  '•ound  th«  sun.  Thus  the  generality  of  our 
language  is  preserved,  and  when,  reft  rring  to  any  particular  central  body, 
we  speak  of  an  exterior  and  an  interior  planet,  we  include  the  cases  in 
which  the  former  is  the  sun  and  the  latter  a  satellite ;  as,  for  example,  in 
the  Lunar  theory.  It  is  a  principle  in  dynamics,  that  the  relative  motions 
of  a  system  of  bodies  inter  se  are  no  way  altered  by  impressing  on  all  of 
them  a  common  motion  >t  motion*,  or  a  cvimmon  force  or  forces  accelera- 
ting or  retarding  them  all  equally  in  oor-. taon  directions,  i.  e.  in  parallel 
lines.  Suppose,  therefore,  we  spply  to  aL  the  three  bodies,  S,  P,  and  M, 
alike,  forces  equal  L  t.hoso  with  which  M  atsd  P  attract  S,  but  in  opposite 
directions.     Then  villi  the  relative  motions  both  of  M  and  P  about  S  be 


ESTIMATION  OP  THB  DISTURBING  FORCE. 


881 


aaaltered ;  but  S,  being  now  urged  by  equal  and  opposite  forces  to  and 
from  both  M  and  P,  will  romain  at  reut.  Lot  us  now  consider  bow  either 
of  the  other  bodies,  as  P,  stands  affected  by  these  newly-introduoed  forces, 
in  addition  to  those  which  before  acted  on  it.  It  is  clear  that  now  P  will 
be  simultaneously  acted  on  by  four  forces ;  firstly,  the  attraction  of  S  in 
the  direction  P  S ;  secondly,  an  additional  force,  in  the  same  direction, 
equal  to  its  attraction  on.  8 ;  thirdly,  the  attraction  of  M  in  the  direction 
F  M ;  and  fourthly,  a  force  parallel  to  M  S,  and  equal  to  M's  attraction 
on  S.  Of  these,  the  two  first,  following  the  same  law  of  the  inverso 
square  of  the  distance  S  P,  may  be  regarded  as  one  force,  precisely  as  if 
the  sum  of  the  masses  of  S  and  P  were  collected  in  S ;  and  in  virtue  of 
their  joint  action,  P  will  describe  an  ellipse  about  S,  except  in  so  fur  as 
that  elliptic  motion  is  disturbed  by  the  other  two  forces.  Thus  we  see 
that  in  this  view  of  the  subject  the  relative  disturbing  force  acting  on  P 
iH  no  longer  the  mere  single  attraction  of  M,  but  a  force  resulting  from 
the  composition  of  that  attraction  with  M's  attraction  on  S  transferred  to 
F  in  a  contrary  direction. 

(611.)  Let  C  P  A  be  part  of  the  relative  orbit  of  the  disturbed,  and  M  B 
of  the  disturbing  body,  their  planes  intersecting  in  the  line  of  nodes  SAB; 


!i  '>  '' 

ix'. 


n 


and  having  to  each  other  the  inclination  expressed  by  the  spherical  angle 
P  A  a.  In  M  P,  produced  if  required,  take  M  N  :  M  S  : :  M  S»  :  IM  P. 
Then,  if  S  M'  be  taken  to  represent,  in  quantity  and  direction,  the  accele- 

'  The  reader  will  be  careful  to  observe  the  order  of  the  letters,  where  forces  are 
represented  by  lines.  M  S  represents  a  force  acting  from  M  towards  S,  S  M  from  S 
towards  M. 


>    \ 


S32 


OUTLINES  OF  ASTRONOMY. 


rative  attraction  of  M  on  S,  M  S  will  represent  in  quantity  and  direction 
the  new  force  applied  to  P,  parallel  to  that  line,  and  N  M  will  represent 
on  the  same  scale  the  accelerative  attraction  of  M  on  P.  Consequently, 
the  disturbing  force  acting  on  P  will  be  the  resultant  of  two  forces  applied 
at  P,  represented  respectively  by  N  M  and  M  S,  which  by  the  laws  of 
dynamics  are  equivalent  to  a  single  force  represented  in  quantity  and 
direction  by  N  S,  hut  having  V  for  its  point  of  application. 

(612.)  The  line  N  S,  is  easily  calculated  by  trigonometry,  when  the 
relative  situations  and  real  distances  of  the  bodies  are  known ;  and  the 
force  expressed  by  that  line  is  directly  comparable  with  the  attractive 
forces  of  S  on  P  by  the  following  proportions,  in  which  M,  S,  represent 
tlie  masses  of  those  bodies  which  are  supposed  to  bo  known,  and  to  which, 
at  equal  distances,  their  attractions  are  proportional : — 

Dis-;-.rbing  force  :  M's  attraction  on  S  : :  N  S  :  S  M; 

M's  attraction  on  S  :  S's  attraction  on  M  : :  M  :  S ; 

S's  attraction  on  M  :  S's  attraction  on  P  : :  S  P* :  S  M* :  by  com- 
pounding which  proportions  wo  collect  as  follows :  — 

Disturbing  force  :  S's  attraction  on  P  : :  M  .  N  S  .  S  P« :  S  .  S  M». 

A  few  numerical  examples  are  subjoined,  exhibiting  the  results  of  this 
calculation  in  particular  cases,  chosen  so  as  to  exemplify  its  application 
under  very  various  circumstances,  throughout  the  planetary  system.  In 
each  case  the  numbers  set  down  express  the  proportion  in  which  the  central 
force  retaining  the  disturbed  body  in  its  elliptic  orbit  exceeds  the  disturb- 
ing force,  to  the  nearest  whole  number.  The  calculation  is  made  for  throe 
positions  of  the  disturbing  body — viz.  at  its  greatest,  its  least,  and  its  mean 
distance  from  the  disturbed. 


I  --I 


Disturbing  body. 

Disturbed  body. 

Ratio  at  the 

greatest  distance 

:  1. 

Ratio  at  the 

mean  distance 

:  1. 

Ratio  at  tho 

least  distance 

:  1. 

The  Sun 

The  Moon 

90 
354 

95683 

255208 

67420 

526 

6433 

20248 

179 

312 

147576 

210245 

66592 

626 

6937 

21579 

89 

128 

63268 

26S33 

6519 

526 

1033 

3065 

•Tunitor 

Saturn 

Jupiter 

The  naTth 

The  Earth 

Uranus 

Venus  .......I....* 

Neptune  ..^ 

Mercurv  .......... 

Neptune 

Jupiter 

Ceres 

Saturn  

Jupiter I 

(613.)  If  the  orbit  of  the  disturbing  body  be  circular,  S  M  is  invariable. 
In  this  case,  N  S  will  continue  to  represent  the  disturbing  force  on  (he 
same  invariable  scale,  whatever  may  be  the  configuration  of  the  three 
bodies  with  respect  to  each  other.     If  the  orbit  of  M  be  but  little  elliptic, 


/ 


ESTIMATION  OF  THE  DISTURBINQ  FORCE. 


833 


the  same  will  be  nearly  the  case.  In  what  follows  throughout  this  chapter, 
except  where  the  contrary  is  expressly  mentioned,  we  shall  neglect  the 
excentricity  of  the  disturbing  orbit. 

(614.)  If  P  be  nearer  to  M  than  S  is,  M  N  is  greater  than  M  F,  and 
N  lies  in  M  P  prolonged,  and  therefore  on  the  opposite  side  of  the  plane 
of  P's  orbit  from  that  on  which  M  is  situated.  The  force  N  S  therefore 
urges  P  towards  M's  plane,  and  towards  a  point  X,  situated  between  S 
and  M,  in  the  line  S  M.  If  the  distance  M  P  be  equal  to  M  S  as  when 
P  is  situated,  suppose,  at  D  or  E,  M  N  is  also  equal  to  M  P  or  M  S,  so 
that  N  coincides  with  P,  and  therefore  X  with  S,  the  disturbing  forces  being 
in  these  cases  directed  towards  the  central  body.  But  if  M  P  bo  greater 
than  MS,  M  N  is  less  than  M  P,  and  N  lies  between  M  and  P,  or  on  the 
same  side  of  the  plane  of  P's  orbit  that  M  is  situated  on.  The  force  N 
S,  therefore,  applied  at  P,  urges  P  towards  the  contrary  side  of  that  plane 
towards  a  point  in  the  line  M  S  produced,  so  that  X  now  shifts  to  the 
farther  side  of  S.  In  all  cases,  the  disturbing  force  is  wholly  effective  in 
the  plane  M  P  S,  in  which  the  three  bodies  lie. 

Fig.  77. 


It  is  very  important  for  the  student  to  fix  distinctly  and  bear  constantly 
in  his  mind  these  relatioils  of  the  disturbing  agency  considered  as  a  sinph 
unresolved  force,  since  their  recollection  will  preserve  him  from  many 
mistakes  in  conceiving  the  mutual  actions  of  the  planets,  &c.  on  each 
other.  For  example,  in  the  figures  here  referred  to,  that  of  Art.  611 
corresponds  to  the  case  of  a  nearer  disturbed  by  a  more  distant  body,  as 
the  earth  by  Jupiter,  or  the  moon  by  the  Sun ;  and  that  of  the  present 
article  to  the  converse  case :  as,  for  instance,  of  Mars  disturbed  by  the 
earth.     Now,  in  this  latter  class  of  cases,  whenever  M  P  is  greater  than 


^B?i 


334 


OUTLINES   OP  ASTRONOMY. 


M  S,  or  S  P,  greater  than  2  S  M,  N  lies  on  the  same  side  of  the  plane 
of  P'a  orbit  with  M,  so  that  N  S,  the  disturbing  force,  contrary  to  what 
might  at  first  be  supposed,  always  urges  the  disturbed  planet  out  of  ihe 
plane  of  its  orbit  towards  the  opposite  side  to  that  on  which  the  disturbing 
planet  lies.  It  will  tend  greatly  to  give  clearness  and  definiteness  to  bis 
ideas  on  the  subject,  if  he  will  trace  out  on  various  suppositions  as  to  the 
relative  magnitude  of  the  disturbing  and  disturbed  orbits  (supposed  to  lie 
in  one  plane)  the  form  of  the  oval  about  M  considered  as  a  fixed  point,  in 
which  the  point  N  lies  when  P  makes  a  complete  revolution  round  S. 

(615.)  Although  it  is  necessary  for  obtaining  in  the  first  instance  a 
clear  conception  of  the  action  of  the  disturbing  force,  to  consider  it  in  thia 
way  as  a  single  force  having  a  definite  direction  in  space  and  a  determinate 
intensity,  yet  as  that  direction  is  continually  varying  with  the  position  of 
N  S,  both  with  respect  to  the  radii  S  P,  S  M,  the  distance  P  M,  and  the 
direction  of  Ps  motion,  it  would  be  impossible,  by  so  considering  it,  to 
attain  clear  views  of  its  dynamical  effect  after  any  considerable  lapse  of 
time,  and  it  therefore  becomes  necessary  to  resolve  it  into  other  equivalent 
forces  acting  in  such  directions  as  shall  admit  of  distinct  and  separate 
consideration.  Now  this  may  be  done  in  several  different  modes.  First, 
we  may  resolve  it  into  three  forces  acting  in  fixed  directions  in  space 
rectangular  to  one  another,  and  by  estimating  its  effect  in  each  of  these 
three  directions  separately,  conclude  the  total  or  joint  effect.  This  is  the 
mode  of  procedure  which  affords  the  readiest  and  most  advantageous 
handle  to  the  problem  of  perturbations  when  taken  up  in  all  its  generality, 
and  is  accordingly  that  resorted  to  by  geometers  of  the  modern  school  in 
all  their  profound  researches  on  the  subject.  Another  mode  consists  in 
resolving  it  also  into  three  rectangular  components,  not,  however,  in  fixed 
directions,  but  in  variable  ones,  viz.  in  the  directions  of  the  lines  N  Q, 
Q  L,  and  L  S,  of  which  L  S  is  in  the  direction  of  the  radius  vector  S  P, 
Q  L  in  a  direction  perpendicular  to  it,  and  in  the  plane  in  which  S  P  and 
a  tangent  to  P's  orbit  at  P  both  lie ;  and  lastly,  N  Q  in  a  direction  per- 
pendicular to  the  plane  in  which  P  is  at  the  instant  moving  about  S. 
The  first  of  these  resolved  portions  we  may  term  the  radial  component 
of  the  disturbing  force,  or  simply  the  radial  disturbing  force;  the  second  the 
transversal;  and  the  third  the  orthogonal}  When  the  disturbed  orbit  is 
one  of  small  excentricity,  the  transversal  component  acts  nearly  in  the 
direction  of  the  tangent  to  P's  orbit  at  P,  and  is  therefore  confounded  with 
that  resolved  component  which  we  shall  presently  describe  (art.  618)  under 

'  This  is  a  term  coined  for  the  occasion.    The  want  of  some  appellation  for  this  com- 
ponent of  the  disturbing  force  is  often  felt. 


'/■  '■■ 


RESOLUTION   OF  THE   DISTURBING  FORCE. 


835 


the  name  of  the  tangential  force.     This  is  the  mode  of  resolving  the 
disturbing  force  followed  by  Newton  and  his  immediate  successor,?. 

(Gl*^)  The  immediate  actions  of  these  components  of  the  disturbing 
force  ari  evidently  independent  of  each  other,  being  rectangular  in  their 
directions  j  and  they  affect  the  movement  of  the  disturbed  body  in  modes 
perfectly  distinct  and  characteristic.  Thus,  the  radial  component,  being 
directed  to  or  from  the  central  body,  has  no  tendency  to  disturb  either 
the  plane  of  P's  orbit,  or  the  equable  description  of  areas  by  P  about  S, 
since  the  law  of  areas  propci  tional  to  the  times  is  not  a  character  of  the 
force  of  gravity  only,  but  holds  good  equally,  whatever  be  the  force  which 
retains  a  body  in  an  orbit,  provided  onh/  its  direction  is  always  towards  a 
fixed  centre.'  Inasmuch,  however,  as  its  law  of  variation  is  not  conform- 
able to  the  simple  law  of  gravity,  it  alters  the  elliptic  form  of  P's  orbit, 
by  directly  affecting  both  its  curvature  and  velocity  at  every  point.  In 
virtue,  therefore,  of  the  action  of  this  disturbing  force,  the  orbit  deviates 
from  the  elliptic  form  by  the  approach  or  recess  of  P  to  or  from  S,  so  that 
the  effect  of  the  perturbations  produced  by  this  part  of  the  disturbing 
force  Mis  wi      ,■  on  the  radius  vector  of  the  disturbed  orbit. 

(617.)  "'-■■:  ;  nsversal  disturbing  force  represented  by  QL,  on  the 
other  hand,  has  no  direct  action  to  draw  P  to  or  from  S.  Its  whole  effi- 
ciency is  directed  to  accelerate  or  retard  P's  motion  in  a  direction  at  right 
angles  to  S  P.  Now  the  area  momentarily  described  by  P  about  &,  is, 
cseteris  paribus,  directly  as  the  velocity  of  P  in  a  direction  perpendicular 
to  S  P.  Whatever  force,  therefore,  increases  this  transverse  velocity  of 
P,  accelerates  the  description  of  areas,  and  vice  versd.  With  the  area 
A  S  P  is  directly  connected,  by  the  nature  of  the  ellipse,  the  angle  ASP 
described  or  to  be  described  by  P  from  a  fixed  line  in  the  plane  of  the 
orbit,  so  that  any  change  in  the  rate  of  description  of  areas  ultimately 
resolves  itself  into  a  change  in  the  amount  of  angular  motion  about  S, 
and  gives  rise  to  a  departure  from  the  elliptic  laws.  Hence  arise  what 
are  called  in  the  perturbational  theory  equations  (i.  e.  changes  or  fluctua- 
tions to  and  fro  about  an  average  quantity)  of  the  mean-  motion  of  the 
disturbed  body. 

(618.)  There  is  yet  another  mode  of  resolving  the  disturbing  force  into 
rectangular  components,  which,  though  not  so  well  adapted  to  the  compu- 
tation of  results,  in  reducing  to  numerical  calculation  the  motions  of  the 
disturbed  body,  is  fitted  to  afford  a  clearer  insight  into  the  nature  of  the 
modifications  which  the  form,  magnitude,  and  situation  of  its  orbit  un- 
dergo in  virtue  of  its  action,  and  which  we  shall  therefore  emnloy  in 
preference.    It  consists  in  estimating  the  components  of  the  disturbing 

*  Newton,  i.  I. 


V       .  1  ft  *ffllK 


',  i 


'J-H. 


336 


OUTLINES   OF  ASTRONOMY. 


force,  which  He  in  the  plane  of  the  orbit,  not  in  the  direction  wo  have 
termed  radial  and  transversal,  i.  e,  in  that  of  the  radius  vector  P  S  and 
perpendicular  to  it,  but  in  the  direction  of  a  tangent  to  the  orbit  at  P, 
and  in  that  of  a  normal  to  the  curve,  and  at  right  angles  to  the  tangent, 
for  which  reason  these  components  may  be  called  the  tamjcntial  and 
normal  disturbing  forces.  When  the  orbit  of  the  disturbed  body  is  cir- 
cular,  or  ncp.'  ,  so,  this  mode  of  resolution  coincides  wi*V  or  differs  but 
little  from  the  former,  but,  when  the  ellipti-jity  is  considerable,  those 
directions  may  deviate  from  the  radial  and  transversal  directions  to  a^y 
extent.  As  in  the  Newtonian  mode  of  resolution,  the  effect  of  the  one 
component  falls  wholly  upon  the  approach  and  recess  of  the  body  P  to 
the  central  body  S,  and  of  the  other  wholly  on  the  rate  of  description  of 
areas  by  P  round  S,  so  in  this  which  we  are  now  considering,  the  direct 
effect  of  the  one  component  (the  normal)  falls  wholly  on  the  curvature  of  tie 
orbit  at  the  point  of  its  action,  increasing  that  curvature  when  the  normal 
force  acts  ir, wards,  or  towards  the  concavity  of  the  orbit,  and  diminishing 
it  when  in  the  opposite  direction ;  while,  on  the  other  hand,  the  tangcatia' 
component  is  directly  effective  on  the  velocity  of  the  disturbed  body,  in- 
creasing or  diminishing  it  according  as  its  direction  conspires  with  or 
opposes  its  motion.  It  is  evident  enough  that  where  the  object  is  to  trace 
simply  the  changes  produced  by  the  disturbing  force,  in  angle  and  distance 
from  the  central  body,  the  former  mode  of  resolution  must  have  the 
advantage  in  perspicuity  of  view  and  applicability  to  calculation.  It  is 
less  obvious,  but  will  abundantly  appear  in  the  sequel  that  the  latter  offers 
peculiar  advantages  in  exhibiting  to  the  eye  and  the  reason  the  momen- 
tary influeuce  of  the  disturbing  force  on  the  elements  of  the  orbit  itself. 

(519.)  Neither  of  the  last  mentioned  pairs  of  the  resolved  portions  of 
the  disturbing  force  tends  to  draw  P  out  of  the  plane  of  its  orbit  PSA. 
But  the  remaining  or  orthogonal  portion  N  Q  acts  directly  and  solely  to 
produce  that  effect.  In  consequence,  under  the  influence  of  this  force,  P 
must  quit  that  plane,  and  (the  same  cause  continuing  in  action)  must 
describe  a  curve  of  double  curvature  as  it  is  called,  no  two  consecutive 
portions  of  which  lie  in  the  same  plane  passing  through  S.  The  effect 
of  this  is  to  produce  a  continual  variation  in  those  elements  of  the  orbit 
of  P  on  which  the  situation  of  its  plane  in  space  depends ;  i.  e.  on  its 
inclination  to  a  fixed  plane,  and  the  position  in  such  a  plane  of  the  node 
or  line  of  its  intersection  therewith.  As  this,  among  all  the  various 
effects  of  perturbation,  is  that  which  is  at  once  the  most  simple  in  its 
conception,  and  the  easiest  to  follow  into  its  remoter  consequences,  we 
aball  begin  with  its  explanation. 

(620.)  Suppose  that  up  to  P  (Art.  611,  614,)  the  body  were  describing 


'•■     y 


EFFECTS   OF  THE   ORTHOGONAL   FORCE. 


337 


an  undisturbed  orbit  C  P.  Tbcn  at  P  it  would  be  moving  in  tlie  direction 
of  a  tangent  P  R  to  the  ellipse  P  A,  which  prolonged  will  intersect  the 
plane  of  M's  orbit  somewhere  in  the  line  of  nodes,  as  at  R.  Now,  at  P, 
let  the  disturbing  force  parallol  to  N  Q  act.  momentarily  on  P ;  then  P 
will  be  deflected  in  the  direction  of  that  force,  and  instead  of  the  arc  P  jp, 
which  it  would  have  described  in  the  next  instant  if  undisturbed,  will 
describe  the  arc  P  q  lying  in  the  state  of  things  represented  in  Art.  611, 
bfllov,  and  in  Art.  614,  above,  P  'p  with  reference  to  the  plane  PSA. 
Tins,  by  this  action  of  the  disturbing  force,  the  plane  of  P's  orbit  will 
have  shifted  its  position  in  space  from  P  S  i?  (an  elementary  portion  of 
th(  old  orbit)  to  P  S  g-,  one  of  the  new.  Now  the  lines  of  nodes  SAB 
in  >le  former  is  determined  by  prolonging  Pp  into  the  tangent  PR, 
intersecting  the  plane  M  S  B  in  R,  and  joining  S  R.  And  in  like  manner, 
if  we  prolong  P  q  into  the  tangent  P  r,  meeting  the  same  plane  in  r,  and 
join  S  r,  this  will  be  the  new  line  of  nodes.  Thus  we  see  that,  under  the 
circumstances  expressed  in  the  former  figure,  the  momentary  action  of  the 
orthogonal  disturbing  force  will  have  caused  the  line  of  nodes  to  rctro- 
(jrade  upon  the  plane  of  the  orbit  of  the  disturbing  body,  and  under 
those  represented  in  the  latter  to  advance.  And  it  is  evident  that  the 
action  of  the  other  resolved  portions  of  the  disturbing  force  will  not  in 
the  least  interfere  with  this  result,  for  neither  of  them  tends  either  to 
carry  P  out  of  its  former  plane  of  motion,  or  to  prevent  its  quitting  it. 
Their  influence  would  merely  go  to  transfer  the  points  of  intersection  of 
the  tangents  P  p  or  P  g-  from  R  or  r  to  R'  or  /,  points  nearer  to  or  far- 
ther from  S  than  R  r,  but  in  the  same  lines. 

(621.)  Supposing,  now,  M  to  lie  to  the  left  instead  of  the  right  side 
of  the  line  of  nodes  in  fig.  1.,  P  retaining  its  situation,  and  M  P  being 
less  than  M  S,  so  that  X  shuU  still  lie  between  M  and  S.  In  this  situation 
of  things  (or  configuration,  as  it  is  termed  of  the  three  bodies  with 
respect  to  each  other,)  N  will  lie  helow  the  plane  ASP,  and  the  disturb- 
ing force  will  terd  to  raise  the  body  P  above  the  plane,  the  resolved 
orthogonal  portion  N  Q  in  this  case  acting  upwards.  The  disturbed  arc 
Pg  will  therefore  lie  above  V  p,  and  when  prolonged  to  meet  the  plane 
M  S  B,  will  intersect  it  in  a  point  in  advance  of  R ;  so  that  in  this  con- 
figuration the  node  will  advance  upon  the  plane  of  the  orbit  of  M,  pro- 
vided always  that  the  latter  orbit  remains  fixed,  or,  at  least,  does  not  itself 
shift  its  position  in  such  a  direction  as  to  defeat  this  result. 

(622.)  Generally  speaking,  the  node  of  the  disturbed  orbit  will  recede 

upon  any  plane  which  we  may  consider  as  fixed,  whenever  the  action  of 

the  orthogonal  disturbing  force  tends  to  bring  the  disturbed  body  nearer 

to  that  plane :  and  vice  versd.     This  will  be  evident  on  a  mere  inspection 

22 


I'l' 


A-.^li 


M  M 


S88 


OUTLINES  OF  ASTRONOMT. 


of  the  annexed  figure,  in  which  C  A  represents  a  semicircle  of  the  projoo- 
tion  of  the  fixed  plane  as  seen  from  S  on  the  sphere  of  the  heavens,  and 
0  P  A  that  of  the  plane  of  Ps  undisturbed  orbit,  the  motion  of  P  bebg 
in  the  direction  of  the  arrow,  from  C  the  ascending,  to  A  the  descend* 
ing  node.    It  b  at  once  seen,  by  prolonging  Tq^Vq'  into  arcs  of  great 


circles,  P  r,  P  r,  (forwards  or  backwards,  as  the  case  may  be)  to  meet 
C  A,  that  the  node  will  have  retrograded  through  the  arc  A  r,  or  C  r, 
whenever  P  q  lies  between  0  P  A  and  0  A,  or  when  the  perturbing  force 
carries  V  towards  the  fixed  plane,  but  will  have  advanced  through  A  r'  or 
C  r'  when  P  q'  lies  above  G  P  A,  or  when  the  disturbing  impulse  has 
lifted  P  above  its  old  orbit  or  away  from  the  fixed  plane,  and  this  with 
out  any  referenoe  to  tohether  the  undisturbed  orbilual  motion  o  P  at  the 
moment  is  carrying  it  towards  the  plane  0  A  or  from  it,  as  in  the  two 
cases  represented  in  the  figure. 

(623.)  Let  us  now  consider  the  mutual  disturbance  of  two  bodies  M 
and  P,  in  the  various  configurations  in  which  they  may  be  presented  to 
each  other  and  to  their  common  central  body.  And  first,  let  us  take  the 
case,  as  the  simplest,  where  the  disturbed  orbit  is  exterior  to  that -of  the 
disturbing  body  (as  in  fig.  art.  614),  and  the  dbtance  between  the  orbits 
greater  than  the  semiaxis  of  the  smaller.  First,  let  both  planets  lie  on 
the  same  side  <^  the  line  of  nodes.  Then  (as  in  art.  620)  the  direc- 
tion of  the  whole  disturbing  force,  and  therefore  also  that  of  its  ortho- 
gonal component,  will  be  towards  the  opposite  side  of  the  plane  of  Fs 
orbit  from  that  on  which  M  lies.  Its  effect  therefore  will  be,  to  draw  P 
out  of  its  plane  in  a  direction  from  the  plane  of  M's  orbit,  so  that  in  this 
state  of  things  the  node  will  advance  on  the  latter  plane,  however  P  and 
M  may  be  situated  ir  these  semicircumferences  of  their  respective  orbits. 
Suppose,  next,  M  transferred  to  the  opposite  side  of  the  line  of  nodes, 
then  will  the  direction  of  its  action  on  P,  with  respect  to  the  plahe  of  P's 
orbit,  be  reversed,  and  P  in  quitting  that  plane  will  now  approach  to 
instead  of  receding  from  the  plane  of  M's  orbit,  so  that  its  node  will  now 
recede  on  that  plane. 

(624.)  Thus,  while  M  and  P  revolve  about  S,  and  in  the  course  of  many 
revolutions  of  each  are  presented  to  each  other  and  to  S  in  all  possible 
configurations,  the  node  of  P's  orbit  will  always  advance  on  M's  when 


MOTION  OF  THE  NODES. 


889 


both  bodies  are  on  tbe  same  side  of  the  line  of  nodes,  and  recede  when 
on  the  opposite!.  They  will,  therefore,  on  an  average,  advance  and  recede 
doriDg  eqnal  times  (supposing  the  orbits  nearly  oironlar).  And,  there- 
fore, if  their  advance  were  at  each  instant  of  its  duration  equally  rapid 
wi*h  their  recess  at  each  corresponding  instant  during  that  phase  of  the 
movement,  they  would  merely  oscillate  to  Ukd  fro  about  a  mean  position, 
without  any  permanent  motion  in  either  direction.  But  this  is  not  the 
case.  The  rapidity  of  their  recess  in  every  position  favourable  to  recess 
is  greater  than  that  of  their  advance  in  the  corresponding  opposite  posi- 
tion.   To  show  this,  let  us  consider  any  two  configurations  in  which  M's 

Fig.  79. 


H: 


'M'.^Rt|T 


f't^y'w*  -•  '■'( ; 


phases  are  diametrically  opposite,  so  that  the  triangles  P  S  M,  P  S  M', 
shall  lie  in  one  plane,  having  any  inclination  to  P's  orbit,  according  to  the 
situation  of  P.  Produce  P  S,  and  draw  M  mj  Wm'  perpendicular  to  it, 
which  will  therefore  be  equal.  Take  M  N  :  M  S  : :  M  S' :  MP*,  and 
M'  F  :  M'  S  : :  M'  S« :  M'  P :  then,  if  the  orbits  be  nearly  circles,  and 
therefore  M  S  =  M'  S,  N'  M'  will  b«  less  than  M  Nj  and  therefore 
(since  P  M'  is  greater  tiian  P  M)  P  N' :  P  M'  in  a  greater  ratio  than 
P  X  :  P  M ;  and  consequently,  by  similar  triangles,  drawing  N  n,  N'  n 
perpendicular  \'o  P  S,  N'  n'  :  M'  m  in  a  greater  ratio  than  N  n  :  M  m,  and 
therefore  N'  n*  is  greater  than  N  n.  Now  the  plane  P  M  M'  intersects 
P's  orbit  in  P  8,  and  being  inclined  to  that  orbit  at  the  same  angle 
through  its  whole  extent,  if  from  N  and  N'  perpendiculars  be  conceived 
let  fall  on  that  orbit,  these  will  be  to  each  other  in  the  proportion  of  N  n, 
^p';  and  therefore  the  perpendicular  from  N'  will  be  greater  than  that 
from  N.  Now  since  by  wet.  611  N'  S  and  N  S  represent  in  quantity  and 
tlirection  the  total  disturbing  forces  of  M'  and  M  on  P  respectively,  there- 
fore these  pei^ndicuiare  express  (art.  615)  the  orthogonal  dbturbing 


IC 


840 


\ 


OUTLINES  OF  ASTRONOMY. 


forces,  the  former  of  which  tends  (as  above  shown)  to  make  the  nodes 
recede,  and  the  latter  to  advance;  and  therefore  the  preponderance  in 
every  such  pair  of  situations  of  M  is  in  favour  of  a  retrograde  motion. 

(625.)  Let  us  next  consider  the  case  where  the  distance  between  the 
orbits  is  less  than  the  semiazis  of  the  interior,  or  in  which  the  least  di». 
tance  of  M  from  P  b  less  than  M  S.    Take  any  situation  of  P  with 


^■:*ii  *     >»■  >»  t     fill,     »^*  v*«     -o  »f «  (. 


..:.......  u...   i-jg,  80. 


■'■"^»«-  '  "j    ;,:<•■*     \     '. 


.1/.  . 


respect  to  the  line  of  nodes  AC.  Then  two  points  d  and  e,  distant  by 
less  than  120",  can  be  token  on  the  orbit  of  M  equidistant  from  P  mth 
S.  Suppose  M  to  occupy  successively  every  possible  situation  in  its  orbit, 
P  retaining  its  place ;  —  then,  if  it  were  not  for  the  existence  of  the  arc 
de,  in  which  the  relations  of  art.  624  are  reversed,  it  would  appear  by 
the  reasoning  of  that  article  that  the  motion  of  the  node  ia  direct  wlica 
M  occupies  any  part  of  the  semiorbit  F  M  B,  and  retrograde  when  it  is  in 
the  opposite,  but  that  the  retrograde  motion  on  the  whole  would  predom- 
inate. Much  more  then  will  it  predominate  when  there  exists  an  arc 
d'Me  within  which  if  M  be  placed,  its  action  will  produce  a  retrograde 
instead  of  a  direct  motion.  -  ■-       ^  -..  ^i  ■  ,..,, 

(626.)  This  supposes  that  the  arc  de  lies  wholly  in  the  semicircio 
FeZB.  But  suppose  it  to  lie,  as  in  the  annexed  figure,  partly  within  and 
partly  without  that  circle.     The  greater  part  d  B  necessarily  lies  within 


r>    /C  .ft/-     ,. 


hi^/hyr-i'-j'  fyit  s;|^;t;i'/;y:F-jp,i'; 


■V  .  f^ ' 


it,  and  not  only  so,  but  within  that  portion,  the  point  of  M's  orbit  nearest 
to  P,  in  which,  therefore,  the  retrograding  force  has  its  maximum,  is  sit- 
uated. Although,  therefore,  in  the  portion  Be,  it  is  true,  the  retrograde 
t<indency  otherwise  general  over  the  whole  of  thai  semicircio  (Art.  624) 


Hi 


!t 


MOTION  OF  THE  NODES. 


841 


will  be  revorBod,  yet  the  effect  of  this  ^11  be  much  more  than  counter- 
balanced by  the  more  energetic  and  more  prolonged  retrograde  actio;:  over 
dB]  and,  therefore,  in  this  case  also,  on  the  average  of  every  possible 
situation  of  M,  the  motion  of  the  node  will  be  retrograde.  "^'^ 

(627.)  Let  us  lastly  consider  an  interior  planet  disturbed  by  an  exte- 
rior. Take  M  D  and  M  E  (fig.  of  art.  611.)  each  equal  to  M  S.  Then 
first,  when  P  is  between  D  and  the  node  A,  being  nearer  than  S  to  M, 
the  disturbing  force  acts  towards  M's  orbit  on  the  side  on  which  M  lies, 
and  the  node  recedes.  It  also  recedes  when  (M  retaining  the  same  situa- 
tion) P  is  in  any  part  of  the  arc  E  0  from  E  to  the  other  i^ode,  because 
iu  that  situation  the  direction  of  the  disturbing  force,  it  is  true,  is  re- 
versed, but  that  portion  of  P's  orbit  being  also  reversely  situated  with 
respect  to  the  plane  of  M's,  P  is  still  urged  towards  the  latter  plane,  but 
on  the  side  opposite  to  M.  Thus,  (M  holding  its  place)  whenever  P  is 
anywhere  in  DA  or  EG,  the  node  recedes.  On  the  other  hand,  it  ad- 
vances whenever  P  is  between  A  and  E  or  between  0  and  D,  because,  in 
these  arcs,  only  one  of  the  two  determining  elements  (viz.  the  direction 
of  the  disturbing  force  with  respect  to  the  plane  of  P's  orbit;  and  the 
situation  of  the  one  plane  with  respect  to  the  other  as  to  above  and  be- 
low) has  undergone  reversal.  Now  first,  whenever  M  is  anywhere  but  in 
the  line  of  nodes,  the  sum  of  the  arcs  D  A  and  E  C  exceeds  a  semicircle, 
and  that  the  more,  the  nearer  M  is  to  a  position  at  right  angles  to  the 
line  of  nodes.  Secondly,  the  arcs  favourable  to  the  recess  of  the  node 
comprehend  those  situations  in  which  the  orthogonal  disturbing  force  is 
most  powerful,  and  vice  versd.  This  is  evident,  because  as  P  approaches 
D  or  E,  this  component  decreases,  and  vanishes  at  those  points  (612.) 
The  movement  of  the  node  itself  also  vanishes  when  P  comes  to  the 
node,  for  although  in  this  position  the  disturbing  orthogonal  force  neither 
vanishes  nor  changes  its  direction,  yet,  since  at  the  instant  of  P's  passing 
the  node  (A)  the  recess  of  the  node  is  changed  into  an  advance,  it  must 
necessarily  at  that  point  be  stationary.'  Owing  to  both  these  causes, 
therefore,  (that  the  mode  recedes  during  a  longer  time  than  it  advances, 
and  that  a  more  energetic  force  acting  in  its  recess  causes  it  to  recede 
more  rapidly,)  the  retrograde  motion  will  preponderate  on  the  whole  in 
each  complete  synodic  revolution  of  P.  And  it  is  evident  that  the  rea- 
soning of  this  and  the  foregoing  articles,  is  no  way  vitiated  by  a  moderate 
amount  of  excentricity  in  either  orbit. 

It  would  seem,  at  first  sight,  as  if  a  change  per  ialtum  took  place  here,  but  the 
continuity  of  the  node's  motion  will  be  apparent  from  an  inspection  of  the  annexed 
figure,  where  6 ad  is  a  portion  of  P's  disturbed  path  near  the  node  A,  concave  towards 
the  plane  G  A.    The  momentary  place  of  the  moving  node  is  determined  by  the  tnter< 


\V 


842 


OUTLINES  OF  ASTRONOMY. 


(628.)  It  18  therefore  a  general  proposition,  that  on  the  avera^  t»*  «ach 
complete  synodio  revolution,  the  node  of  every  disturbed  planet  ret^des 
upon  the  orbit  of  the  disturbing  one,  or  in  other  wordj,  that  in  every  pair 
of  orbits,  the  node  of  each  recedes  upon  the  other,  and  of  course  upon  any 
intermediate  plane  which  we  may  regard  as  fixed.  On  a  plane  not  inter- 
mediate b|^tween  them,  however,  the  node  of  one  orbit  will  advance,  and 
that  of  the  other  will  recede.    Suppose  for  instance,  C  A  0  to  be  a  plane 


ij  '/ni/h^'i- 


*N  ■  ■  ,<'r4'- 


intermediate  between  PP  and  M  M  the  two  orbits.  1{  pp  and  mm  be 
the  new  positions  of  the  orbits,  the  node  of  P  on  M  will  have  receded 
from  A  to  5,  that  of  M  on  P  from  A  to  4,  that  of  P  and  M  on  C  G  re- 
spectively from  A  to  1  and  from  A  to  2.  But  if  F  A  F  be  a  plane  not 
intermediate,  the  node  of  M  on  that  plane  has  receded  from  A  to  6,  but 
that  of  P  will  have  advanced  from  A  to  7.  If  the  fixed  planer  have  not 
a  common  intersection  with  those  of  both  orbits,  it  is  equally  easy  to  see 
that  the  node  of  the  disturbed  orbit  may  either  recede  on  both  that  plane 
and  the  disturbing  orbit,  or  advance  on  the  one  and  recede  on  the  other, 
according  to  the  relative  situation  of  the  planes. 

(629.)  This  is  the  case  with  the  planetary  orbits.    They  do  not  all 

section  of  the  tangent  (  e  with  A  G,  which  as  5  passes  through  a  to  d,  recedes  from  A 


to  a,  rests  there  for  an  instant,  and  then  advances  again. 


MOTION  OF  THE  NODES. 


843 


intersect  each  other  in  a  common  node.  Although  perfectly  true,  there- 
fore, that  the  node  of  any  one  planet  would  recede  on  the  orbit  of  any  and 
each  other  by  the  individual  action  of  that  other,  yet,  when  all  act  to- 
gether, recess  on  one  plane  may  be  equivalent  to  advance  on  another,  so 
that  the  motion  of  th(  uode  of  any  one  orbit  on  a  given  plane,  arising  from 
their  joint  action,  taking  into  account  the  diiferent  situations  of  all  the 
planes,  becomes  a  curiously  complicated  phsenomenon  whose  law  cannot 
be  very  easily  expressed  in  words,  though  reducible  to  strict  numerical 
statement,  being,  in  fact,  a  mere  geometrical  result  of  what  is  above 
shown.  .  sty  <■>':, jti'iVi- -,H  (  t^'vir^:^  :;'  >>,■-;■■-,;•..', ■ 

(630.)  The  nodes  of  all  the  planetary  orbits  on  the  true  ecliptic,  as  a 
matter  of  fact,  are  retrograde,  though  they  are  not  all  so  on  a  fixed  plane, 
such  as  we  may  conceive  to  exist  in  the  planetary  system,  and  to  be  a 
plane  of  reference  unaffected  by  their  mutual  disturbances.  It  is,  how- 
ever, to  the  ecliptic,  that  we  are  under  the  necessity  of  referring  their 
movements  from  our  station  in  the  syntcm ;  and  if  we  would  transfer  our 
ideas  to  a  fixed  plane,  it  becomes  necessary  to  take  account  of  the  varia- 
tion of  the  ecliptic  itself,  produced  by  the  joint  action  of  all  the  planets. 

(631.)  Owing  to  the  smallness  of  the  masses  of  the  planets,  and  their 
great  distances  from  each  other,  the  revolutions  of  their  nodes  are  exces- 
sively slow,  being  in  every  case  less  than  a  single  degree  per  century,  and 
in  most  cases  not  amounting  to  half  that  quantity.  It  is  otherwise  with 
the  moon,  and  that  owing  to  two  distinct  reasons.  First,  that  the  disturb- 
ing force  itself  arising  from  the  sun'e  action,  (as  appears  from  the  table 
given  in  art.  612,)  bears  a  much  larger  proportion  to  the  earth's  central 
attraction  on  the  moon  than  in  the  case  of  any  planet  disturbed  by  any 
other.  And  secondly,  because  the  synodic  revolution  of  the  moon, 
within  which  the  average  is  struck  (and  always  on  the  side  of  recess),  is 
only  29  J^  days,  a  period  much  shorter  than  that  of  any  of  the  planets, 
and  vastly  so  than  that  of  several  among  them.  All  this  is  agreeable  to 
what  has  already  been  stated  (art.  407,  408,)  respecting  the  motion  of  the 
moon's  nodes,  and  it  is  hardly  necessary  to  mention  that,  when  calculated, 
as  it  has  been,  d,  priori,  from  an  exact  estimation  of  all  the  acting  forces, 
the  result  is  found  to  coincide  with  perfect  precision  with  that  immediately 
derived  from  observation,  so  that  not  a  doubt  can  subsist  as  to  this  being 
the  real  process  by  which  so  remarkable  an  effect  is  produced. 

(632.)  So  far  as  the  physical  condition  of  each  planet  is  concerned,  it 
is  evident  that  the  position  of  their  nodes  can  be  of  little  importance.  It 
is  otherwise  with  the  mutual  inclinations  of  their  orbits  with  respect  to 
each  other,  and  to  the  equator  of  each.  A  variation  in  the  position  of  the 
ecliptic,  for  instance,  by  which  its  pole  should  shift  its  distance  from  the 


If 


I 


m 


844 


OUTLINES  OF  A8TR0N0MT. 


pole  of  the  equator,  would  disturb  our  aeasoDS.  Should  the  plane  of  the 
curth's  orbis,  for  instauce,  ever  be  so  changed  as  to  bring  tho  ccliptio  to 
coincido  with  the  equator,  we  should  have  perpetual  spring  over  all  the 
world ;  and  on  the  other  hand,  should  it  coincide  with  a  meridian,  the 
extremes  of  summer  and  winter  would  become  intolerable.  The  inquiry, 
then,  of  the  variations  of  inclination  of  the  planetary  orbits  inter  te,  i» 
one  of  much  higher  practical  interest  than  thoao  of  their  nodes. 
U  (083.)  Referring  to  tho  figures  of  art.  610,  et  seq.,  it  is  evident  that 
the  plane  S  Vq,  in  which  tho  disturbed  body  moves  during  an  instant  of 
time  from  its  quitting  P,  is  differently  inclined  to  the  orbit  of  M,  or  to  a 
fixed  plane,  from  the  original  or  undisturbed  plane  P  S^^.  The  difference 
of  absolute  position  of  these  two  planes  in  space  is  the  angle  between  the 
planes  P  S  R  and  P  S  r,  and  is  therefore  calculable  by  spherical  trigono- 
metry, when  the  angle  R  S  r  or  the  momentary  recess  of  tho  node  is 
known,  and  also  the  inclination  of  the  planes  of  the  orbits  to  each  other. 
We  perceive,  then,  that  between  tho  momentary  chango  of  inclination, 
and  the  momontary  recess  of  the  node,  there  exists  an  intimate  relatioi, 
and  that  the  research  of  the  one  is  in  fact  bound  up  in  that  of  tho  other. 
This  may  be,  perhaps,  made  clearer,  by  considering  the  orbit  of  P  to  be 
not  merely  an  imaginary  line,  but  an  actual  circle  or  elliptic  hoop  of  some 
rigid  material,  without  inertia,  on  which,  as  on  a  wire,  the  body  P  may 
slide  as  a  bead.  It  is  evident  that  the  position  of  this  hoop  will  be  deter- 
mined at  any  instant,  by  its  inclination  to  the  ground  plane  to  which  it 
is  referred,  and  by  the  place  of  its  intersection  therewith,  or  node.  It 
will  also  be  determined  by  the  momentary  direction  of  P's  motion,  which 
(having  no  inertia)  it  must  obey ;  and  any  change  by  which  P  should,  in 
the  next  instant,  alter  its  orbit,  would  be  equivalent  to  a  shifting,  bodily, 
of  the  whole  hoop,  changing  at  once  its  inclination  and  nodes. 

(634.)  One  immediate  conclusion  from  what  has  been  pointed  out 
above,  is  that  where  the  orbits,  as  in  the  case  of  the  planetary  system  and 
the  moon,  are  slightly  inclined  to  one  another,  the  momcnuiry  variations 
of  the  inclination  are  of  an  order  much  inferior  in  magnitude  to  those  in 
the  place  of  the  node.  This  is  evident  on  a  mere  inspection  of  our  figure, 
ihe  angle  R  P  r  being,  by  reason  of  the  small  inclination  of  the  planes 
S  P  R  and  R  S  r,  necessarily  much  smaller  than  the  angle  R  S  n  In  pro- 
portion as  the  planes  of  the  orbits  are  brought  to  coincidence,  a  very  tri- 
fling angular  movement  of  Pp  about  P  S  as  an  axis  will  make  a  great 
variation  in  the  situation  of  the  point  r,  where  its  prolongation  intersects 
the  ground  plane. 

(635.)  Referring  to  the  figure  of  art.  622,  we  perceive  that  although 
the  motion  of  the  node  is  retrograde  whenever  the  momentary  disturbed 


.  I 


CHANQB  OF  INCLINATION. 


845 


arc  P  Q  lies  botwoon  the  pianos  C  A  and  0  0  A  of  the  two  orbits,  and 
vice  versd,  indifferently  whether  P  be  in  the  act  of  receding  from  the 
plane  G  A,  a8  in  the  quadrant  C  G,  or  of  approaching  to  it,  as  in  G  A, 
yet  the  same  identity  as  to  the  character  of  the  change  does  not  subsist 
in  respect  of  the  inclination.  The  inclination  of  the  disturbed  orbit  (t.  e. 
of  its  momentary  element)  Pg  or  Pj',  is  measured  by  the  spherical  angle 
Pr  H  or  P /  H.  Now  in  the  quadrant  C  G,  P  r  H  is  less,  and  P  r'  H 
greater  than  P  C  H ;  but  in  G  A,  the  converse.  Hence  this  rule  : — 1st, 
If  the  disturbing  force  urge  P  towards  the  plane  of  M's  orbit,  and  the 
undisturbed  motion  of  P  carry  it  also  towards  that  plane;  and  2dly,  if  the 
disturbing  force  urge  P  from  that  plane,  while  P's  undisturbed  motion  also 
carries  it  from  it,  in  either  case  the  inclination  momentarily  inc  ^cases ;  but 
if,  Sdly,  the  disturbing  force  act  to,  and  P's  motion  carry  it  tiom — or  if 
the  force  act  from,  and  the  motion  carry  it  to,  that  plane,  the  inclination 
momenturily  diminishes.  Or  (including  all  the  cases  under  one  al  3rnative^ 
if  the  action  of  the  disturbing  force  and  the  undisturbed  motion  of  P  with 
reference  to  the  plane  of  M's  orbit  be  of  the  same  character,  the  inclina- 
tion increases ;  if  of  contrary  characters,  it  diminishes. 

(636.)  To  pass  from  the  momentary  changes  which  take  place  in  'bo 
relations  of  nature  to  the  accumulated  effects  produced  in  considera  jle 
lapses  of  time  by  the  continued  action  of  the  same  causes,  under  circum- 
stances varied  by  these  very  effects,  is  the  business  of  the  integral  calculus. 
Without  going  into  any  calculations,  however,  it  will  be  easy  for  us  to 
demonstrate  from  the  principles  above  laid  down,  the  leading  features  of 
this  part  of  the  planetary  theory,  viz.  the  periodic  nature  of  the  change 
of  the  inclinations  of  two  orbits  to  each  other,  the  re-establishment  of  their 
original  values,  and  the  consequent  oscillation  of  each  plane  about  a  certain 
mean  position.  As  in  explaining  the  motion  of  the  nodes,  we  will  com« 
mence,  as  the  simplest  case,  with  that  of  an  exterior  planet  disturbed  by 
an  interior  one  at  less  than  half  its  distance  from  the  cen . .-..'•  body.  Let 
A  C  A'  be  the  great  circle  of  the  heavens  into  which  M's  oiDit  seen  from 
S  is  projected,  extended  into  a  straight  line,  and  AgQhA'  the  corre- 
sponding projection  of  the  orbit  of  P  so  seen.  Let  ''I  occupy  some  fixed 
situation,  suppose  in  the  semicircle  A  C,  and  let  I'  describe  a  complete 
revolution  from  A  through  ^^  C  ^  to  A'.  Then  while  it  is  between  A  and 
g  or  in  its  first  quadrant,  its  motion  is  from  the  plane  of  M's  orbit,  and 
at  the  same  time  the  orthogonal  force  acts  from  that  plane :  the  inclina- 
tion, therefore,  (art.  635)  increases.  In  the  second  quadrant  the  motion 
of  P  is  to,  but  the  force  continues  to  act  from,  the  plane,  and  the  inclina- 
tion  again  decreases.  A  similar  alternation  takes  place  in  its  course 
through  the  quadrants  C  A  and  hA.     Thus  the  plane  of  P's  orbit  osoil- 


'1. 


346 


OUTLINES  OP  ASTRONOMY. 


Fig.  84. 


lates  to  and  fro  about  its  mean  position  twice  in  each  revolution  of  P. 
During  this  process  if  M  held  a  fixed  position  at  Q,  the  forces  being 
symmetrically  alike  on  either  side,  the  extent  of  these  oscillations  would  be 
exactly  equal,  and  the  inclination  at  the  end  of  one  revolution  of  P  would 
revert  precisely  to  its  original  value.  But  if  M  be  elsewhere,  this  will 
not  be  the  case,  and  in  a  single  revolution  of  P,  only  a  partial  compensa- 
tion will  be  operated,  and  an  overplus  on  the  side,  suppose  of  diminution, 
will  remain  outstanding.  But  when  M  comes  to  M',  a  point  equidistant 
from  G  on  the  other  side,  this  eflFect  will  be  precisely  reversed  (supposing 
the  orbits  circular).  On  the  average  of  both  situations,  therefore,  the 
effect  will  be  the  same  as  if  M  were  divided  into  two  equal  portions,  one 
placed  at  M  and  the  other  at  M',  which  will  annihilate  the  preponderance 
in  question  and  effect  a  perfect  restoration.  And  on  an  average  of  all 
possible  situations  of  M,  the  effect  will  in  like  manner  be  the  same  as  if 
its  mass  were  distributed  over  the  whole  circumference  of  its  orbit,  forming 
a  ring,  each  portion  of  which  will  exactly  destroy  the  effect  of  that  simi- 
larly situated  on  the  opposite  side  of  the  line  of  nodes. 

(673.)  The  reasoning  is  precisely  similar  for  the  more  complicated 
cases  of  arts.  (625)  and  (627.)  Suppose  that  owing  either  to  tbc 
proximity  of  the  two  orbits,  (in  the  case  of  an  exterior  disturbed  planet) 
or  to  the  disturbed  orbit  being  interior  to  the  disturbing  one,  there  were 
a  larger  or  less  portion,  d  e,  of  P's  orbit  in  which  these  relations  were 
reversed.  Let  M  be  the  position  of  M'  corresponding  to  d  e,  then  taking 
Or  M'=G!-  M,  there  will  be  a  similar  portion  d'  d  bearing  precisely  the 
same  reversed  relation  to  M',  and  therefore,  the  actions  of  M'  M,  will 
equally  neutralize  each  other  in  this  as  in  the  former  state  of  things. 

(638.)  To  operate  a  complete  and  rigorous  compensation,  however,  it 
is  necessary  that  M  should  be  presented  to  P  in  every  possible  configura- 
tion, not  only  with  respect  to  P  itself,  but  to  the  line  of  nodes,  to  the 
position  of  which  line  the  whole  reasoning  bears  reference.  In  the  case 
of  the  moon  for  example,  the  disturbed  body  (the  moon)  revolves  in 
27''-322,  the  disturbing  (the  sun)  in  365*-256,  and  the  line  of  nodes  in 
6793*-391,  numbers  in  proportion  to  each  other  about  as  1  to  13  and  249 
respectively.  Now  in  18  revolutions  of  P,  and  one  of  M,  if  the  node 
remained  fixed,  P  would  have  been  presented  to  M  so  nearly  in  every 


CHANGE   OF  INCLINATION. 


34T 


configuration  as  to  operate  an  almost  exaot  compensation.  But  in  1 
revolution  of  M,  or  13  of  P,  the  node  itself  has  shifted  3^^  or  about  j', 
of  a  revolution,  in  a  direction  opposite  to  the  revolutions  of  M  and  P,  so 
that  although  P  has  been  brought  back  to  the  same  configuration  with 
respect  to  M,  both  are  -^  of  a  revolution  in  advance  of  the  same  configu- 
ration as  respects  the  node.  The  compensation,  therefore,  will  not  be 
exact,  and  to  make  it  so,  this  process  must  be  gone  through  19  times,  at 
the  end  of  which  both  the  bodies  will  be  restored  to  the  same  relative 
position,  not  only  with  respect  to  each  other,  but  to  the  node.  The 
fractional  parts  of  entire  revolutions,  which  in  this  explanation  have  been 
neglected,  are  evidently  no  farther  influential  than  as  rendering  the  com- 
pensation thus  operated  in  a  revolution  of  the  ^node  slightly  inexact,  and 
thus  giving  rise  to  a  compound  period  of  greater  duration,  at  the  end  of 
which  a  compensation  almost  mathematically  rigorous,  will  have  been 
effected. 

(639.)  It  is  clear  then,  that  if  the  orbits  be  circles,  the  lapse  of  a  very 
moderate  number  of  revolutions  of  the  bodies  will  very  nearly,  and  that 
of  a  revolution  of  the  node  almost  exactly,  bring  about  a  perfect  restora- 
tion of  the  inclinations.  If,  however,  we  suppose  the  orbits  excentric,  it 
is  no  less  evident,  owing  to  the  want  of  symmetry  in  the  distribution  of 
the  forces,  that  a  perfect  compensation  will  not  be  effected  either  in  one 
or  in  any  number  of  revolutions  of  P  and  M,  independent  of  the  motion 
of  the  node  itself,  as  there  will  always  be  some  configuration  more  favour- 
able to  either  an  increase  of  inclination  than  its  opposite  is  unfavourable. 
Thus  will  arise  a  change  of  inclination  which,  were  the  nodes  and  apsides 
of  the  orbits  fixed,  would  be  always  progressive  in  one  direction  until 
the  planes  were  brought  to  coincidence.  But,  1st,  half  a  revolution  of 
the  nodes  would  of  itself  reverse  the  direction  of  this  progression  by 
making  the  position  in  question  favour  the  opposite  movement  of  inclina- 
tion; aud,  2dly,  the  planetary  apsides  are  themselves  in  motion  with 
unequal  velocities,  and  thus  the  configuration  whose  influence  destroys 
the  balance,  is,  itself,  always  shifting  its  place  on  the  orbits.  The  varia- 
tions of  inclination  dependent  on  the  excentricities  are  therefore,  like  those 
independent  of  them,  periodical,  and  being,  moreover,  of  an  order  more 
minute  (by  reason  of  the  smallness  of  the  excentricities)  than  the  latter, 
it  is  evident  that  the  total  variation  of  the  planetary  inclinations  must 
fluctuate  within  very  narrow  limits.  Geometers  have  accordingly  demon- 
strated by  an  accurate  analysis  of  all  the  circumstances,  and  an  exact 
estimation  of  the  acting  forces,  that  such  is  the  case ;  and  this  is  what 
is  meant  by  asserting  the  stability  of  the  planetary  system  as  to  the 
mutual  inclinations  of  its  orbits.     By  the  researches  of  Lagrange  (of 


•i'l 


li 


1 1 


.  .  n 


i'--: 


in 


•*:' 


4; 


J- 1*1 


us 


OUTLINES  OF  ASTRONOMY. 


whose  analytical  conduct  it  is  impossible  here  to  give  any  idea,)  the 
following  elegant  theorem  has  been  demonstrated :  — 

"  If  the  muss  of  every  planet  be  multiplied  hy  the  square  root  of  the 
major  axis  of  its  orbit,  and  the  prodiict  by  the  square  of  the  tangent  of 
its  inclination  to  a  fixed  plane,  the  sum  of  all  these  products  will  be  con- 
stantly the  same  under  the  influence  of  their  mutual  attraction."  If  the 
present  situation  of  the  plane  of  the  ecliptic  be  taken  for  that  fixed  plane 
(the  ecliptic  itself  being  variable  like  the  other  orbits),  it  is  found  that 
this  sum  is  actually  very  small :  it  must,  therefore,  always  remain  so. 
This  remarkable  theorem  alone,  then,  would  guarantee  the  stability  of  the 
orbits  of  the  greater  planets ;  but  from  what  has  above  been  shown  of  the 
tendency  of  each  planet  to  work  out  a  compensation  on  every  other,  it  is 
evident  that  the  minor  ones  are  not  excluded  from  this  beneficial  arrange- 
ment.        .'■■!,■■       •     .  ■•      -  -       ■  ••     '      •     :    "   •      •',■•• 

(640.)  Meanwhile,  there  is  no  doubt  that  the  plane  of  the  ecliptic  does 
actually  vary  by  the  actions  of  the  planets.  The  amount  of  this  variation 
is  about  48"  per  century,  and  has  long  been  recognized  by  astronome-s, 
by  an  increase  of  the  latitudes  of  all  the  stars  in  certain  situations,  and 
their  diminution  in  the  opposite  regions.  Its  eflfect  is  to  bring  the  ecliptic 
by  so  much  per  annum  nearer  to  coincidence  with  the  equator ;  but  from 
what  we  have  above  seen,  this  diminution  of  the  obliquity  of  the  ecliptic 
will  not  go  on  beyond  certain  very  moderate  limits,  after  which  (although 
in  an  immense  period  of  ages,  being  a  compound  cycle  resulting  from  the 
joint  action  of  all  the  planets,)  it  will  again  increase,  and  thus  oscillate 
backward  and  forward  about  a  mean  position,  the  extent  of  its  deviation 
to  one  side  and  the  other  being  less  than  1°  21'. 

(641.)  One  effect  of  this  variation  of  the  plane  of  the  ecliptic, — that 
which  causes  its  nodes  on  a  fixed  plane  to  change, — is  mixed  up  with  the 
precession  of  the  equinoxes,  and  undistinguishable  from  it,  except  in 
theory.  This  last-mentioned  phsenomenon  is,  however,  due  to  another 
cause,  analogous,  it  is  true,  in  a  general  point  of  view,  to  those  above 
eonsidered,  but  singularly  modified  by  the  circumstances  under  which  it 
is  produced.  We  sL.U  endeavour  to  render  these  modifications  intelli- 
gible, as  far  as  they  can  be  made  so  without  the  intervention  of  analytical 
formulae. 

(642.)  The  precession  of  the  equinoxes,  as  we  have  shown  in  art.  312, 
consists  in  a  continual  retrogradation  of  the  node  of  the  earth's  equator  on 
the  ecliptic  j  and  is,  therefore,  obviously  an  effect  so  far  analogous  to  the 
general  phaenomenon  of  the  retrogradation  of  the  nodes  of  the  orbits  on 
each  other.  The  immense  distance  of  the  planets,  however,  compared 
with  the  size  of  the  earth,  and  the  smallness  of  their  masses  compared  to 


PRECESSION  OF  THE  EQUINOXES. 


349 


that  of  the  sun,  puts  ilieir  action  out  of  the  question  in  the  inquiry  of  its 
cause,  and  we  must,  therefore,  look  to  the  massive  though  distant  sun, 
and  to  our  near  though  minute  neighbour,  the  moon,  for  its  explanation. 
This  will,  accordingly,  be  found  in  their  disturbing  action  on  the  redun- 
dant matter  accumulated  on  the  equator  of  the  earth,  by  which  its  figure 
is  rendered  spheroidal,  combined  with  the  earth's  rotation  on  its  axis.  It 
is  to  the  sagacity  of  Newton  that  we  owe  the  discovery  of  this  singular 
mode  of  action. 

(643.)  Suppose  in  our  figure  (art.  611,)  that  instead  of  one  body,  P, 
revolving  round  S,  there  were  a  succession  of  particles  not  coherent,  but 
forming  a  kind  of  fluid  ring,  free  to  change  its  form  by  any  force  applied. 
Then,  while  this  ring  revolved  round  S  in  its  own  plane,  under  the  dis- 
turbing influence  of  the  distant  body  M,  (which  now  represents  the  moon 
or  the  sun,  as  P  does  one  of  the  particles  of  the  earth's  equator,)  two 
things  would  happen :  1st,  its  figure  would  be  bent  out  of  a  plane  into  an 
undulated  form,  those  parts  of  it  within  the  arcs  D  A  and  E  C  being  ren- 
dered more  inclined  to  the  plane  of  M's  orbit,  and  those  within  the  arcs 
AE,  CD,  less  so  than  they  would  otherwise  be;  2dly,  the  nodes  of  this 
ring,  regarded  as  a  whole,  without  respect  to  its  change  of  figure,  would 
retreat  upon  that  plane. 

(644.)  But  suppose  this  ring,  instead  of  consisting  of  discrete  mole- 
cules free  to  move  independently,  to  be  rigid  and  incapable  of  such  flexure, 
like  the  lioop  we  have  supposed  in  art.  633,  but  having  inertia,  then  it  is 
evident  that  the  effort  of  those  parts  of  it  which  tend  to  become  more 
inclined  will  act  through  the  medium  of  the  ring  itself  (as  a  mechanical 
engine  or  lover)  to  counteract  the  effort  of  those  which  have  at  the  same 
instant  a  contrary  tendency.  In  so  far  only,  then,  as  there  exists  an  excess 
on  the  one  or  the  other  side  will  the  inclination  change,  an  average  being 
struck  at  every  moment  of  the  ring's  motion;  just  as  was  shown  to 
happen  in  the  view  we  have  taken  of  the  inclinations,  in  every  complete 
revolution  of  a  single  disturbed  body,  under  the  influence  of  a  fixed  dis- 
turbing one. 

(645.)  Meanwhile,  however,  the  nodes  of  the  rigid  ring  will  retrograde, 
the  general  or  average  tendency  of  the  nodes  of  every  molecule  being  to 
do  so.  Here,  as  in  the  other  case,  a  struggle  will  take  place  by  the  coun- 
teracting e&';ts  of  the  molecules  contrarily  disposed,  propagated  through 
the  solid  substance  of  the  ring ;  and  thus  at  every  instant  of  time,  an 
average  will  be  struck,  which  being  identical  in  its  nature  with  that 
effected  in  the  complete  revolution  of  a  single  disturbed  body,  will,  in 
every  case,  be  in  favour  of  a  recess  of  the  node,  save  only  when  the  dis- 


850 


OUTLINES   OF  ASTRONOMY. 


I 


turbing  body,  be  it  sun  or  moon,  is  situated  in  the  plane  of  tue  earth's 
equator. 

(646.)  This  reasoning  is  evidently  independent  of  any  consideration 
of  the  cause  which  maintains  the  rotation  of  the  ring ;  whether  the  par- 
ticles be  small  satellites  retained  in  circular  orbits  under  the  equilibrated 
action  of  attractive  and  centr  fugal  forces,  or  whether  they  be  small  masses 
conceived  as  attached  to  a  bet  cf  imaginary  spokes,  as  of  a  wheel,  center- 
ing iu  S,  and  free  only  to  ^hift^  their  planes  by  a  motion  of  those  spokes 
perpendicular  to  the  plane  of  the  wheel.  This  makes  no  difference  in 
the  general  effect );  though  the  different  velocities  of  rotation,  which  may 
be  impressed  on  such  a  system,  may  and  will  have  a  very  great  influence 
both  on  the  absolute  and  relative  magnitudes  of  the  two  effects  in  ques- 
tion —  the  motion  of  the  nodes  and  change  of  inclination.  This  will  be 
easily  understood,  if  we  suppose  the  ring  voithout  a  rotatory  motion,  in  which 
extreme  case  it  is  obvious  that  so  long  as  M  remained  fixed  there  would 
take  place  no  recess  of  nodes  at  all,  but  only  a  tendency  of  the  ring  to  tilt 
its  plane  round  a  diameter  perpendicular  to  the  position  of  M,  bringiitig  it 
towards  the  line  S  M. 

(647.)  The  motion  of  such  a  ring,  then,  as  we  have  been  considering, 
would  imitate,  so  far  as  the  recess  of  the  nodes  goes,  the  precession  of  the 
equinoxes,  only  that  its  nodes  would  retrograde  far  more  rapidly  than  the 
observed  precession,  which  is  excessively  slow.  But  now  conceive  this 
ring  to  be  loaded  with  a  spherical  mass  enormously  heavier  than  itself, 
placed  concentrically  within  it,  and  cohering  firmly  to  it,  but  indifferent, 
or  very  nearly  so,  to  any  such  cause  of  motion ;  and  suppose,  moreover, 
that,  instead  of  one  such  ring  there  are  a  vast  multitude  heaped  together 
around  the  equator  of  such  a  globe,  so  as  to  form  an  elliptical  protube- 
rance, enveloping  it  like  a  shell  on  all  sides,  but  whose  mass,  taken  toge- 
ther, should  form  but  a  very  minute  fraction  of  the  whole  spheroid.  We 
have  now  before  us  a  tolerable  representation  of  the  case  of  nature;'  and 
it  is  evident  that  the  rings,  having  to  drag  round  with  them  in  their  nodal 

*  That  a  perfect  sphere  would  be  bo  inert  and  indifferent  as  to  a  revolution  of  the 
nodes  of  its  equator  under  the  influence  of  a  distant  attracting  body  appears  from  this, 
—  that  the  direction  of  the  resultant  attraction  of  such  a  body,  or  of  that  single  force 
which,  opposed,  would  neutralize  and  destroy  its  whole  action,  is  necessarily  in  a  line 
passing  through  the  centre  of  the  sphere,  and,  therefore,  can  have  no  tendency  to  turn 
the  sphere  one  way  or  other.  It  may  be  objected  by  the  reader,  that  the  whole  sphere 
may  be  conceived  as  consisting  of  ringij  parallel  to  its  equator,  of  every  possible  dia- 
meter, and  that,  therefore,  its  nodes  should  retrograde  even  without  a  protuberant 
equator.  The  inference  is  incorrect,  but  our  limits  will  not  allow  us  to  go  into  an  ex- 
position of  the  fallacy.  We  should,  however,  caution  him,  generally,  that  no  dyna- 
mical subject  is  open  to  more  mistakes  of  this  kind,  which  nothing  but  the  closest 
attention,  in  every  varied  point  of  view,  will  detect. 


V<tf 


fMi.^ 


-t:^,: 


PRECESSION  AND   NUTATION   EXPLAINED. 


351 


revolution  this  great  inert  mass,  will  bavo  their  velocity  of  retrogradation 
proportionally  diminished.  Thus,  then,  it  is  easy  to  conceive  how  a  mo- 
tion similar  to  the  precession  of  the  equinoxes,  and,  like  it,  characterized 
by  extreme  slowness,  will  arise  from  the  causes  in  action. 

(648.)  Now  a  recess  of  the  node  of  the  earth's  equator,  upon  a  given 
plane,  corresponds  to  a  conical  motion  of  its  axis  round  a  perpendicular  to 
that  plane.  But  in  the  case  before  us,  that  plane  is  not  the  ecliptic,  but 
the  moon's  orbit  for  the  time  being;  and  it  may  be  asked  how  we  are  to 
reconcile  this  with  what  is  stated  in  art.  317  respecting  the  nature  of  the 
motion  in  question.  To  this  we  reply,  that  the  nodes  of  the  lunar  orbit, 
beir:  ~  in  a  state  of  continual  and  rapid  retrogradation,  while  its  inclination 
is  preserved  nearly  invariable,  the  point  in  the  sphere  of  the  heavens  round 
which  the  pole  of  the  earth's  equator  revolves  (with  that  extreme  slow- 
ness characteristic  of  the  precession)  is  itself  in  a  state  of  continual  circu- 
lation round  the  pole  of  the  ecliptic,  with  that  much  more  rapid  motion 
which  belongs  to  the  lunar  node.    A  glance  at  the  annexed  figure  will 


in 

!     f 

S)' 


!>!; 


i  I    <    ' 


>t*s 


£  Oi 


explain  this  better  than  words.  P  is  the  pole  of  the  ecliptic,  A  the  pole 
of  the  moon's  orbit,  moving  round  the  small  circle  A  B  0  D  in  19  years ; 
a  the  pole  of  the  earth's  equator,  which  at  each  moment  of  its  progress 
has  a  direction  perpendicular  to  the  varying  position  of  the  line  A  a,  and 
a  velocity  depending  on  the  varying  intensity  of  the  acting  causes  during 
the  period  of  the  nodes.  This  velocity  however  being  extremely  small, 
when  A  comes  to  B,  C,  D,  E,  the  line  A  a  will  have  taken  up  the  posi- 
tions B  6,  C  c,  D  (/,  E  e,  and  the  earth's  p^le  a  will  thus,  in  one  tropical 
revolution  of  the  node,  have  arrived  at  e,  having  described  not  an  exactly 
circular  arc  a  e,  but  a  single  undulation  of  a  wave-shape  or  epicycloidal 
curve,  ah  c  d  e,  with  a  velocity  alternately  greater  and  less  than  its  mean 


852 


Oi;TLr.\^ES   OF  ASTRONOMY. 


motion,  and  this  will  be  repeated  in  every  succeeding  revolution  of  the 
node. 

(649.)  Now  this  is  precisely  the  kind  of  motion  which,  aa  we  hara 
seen  in  art.  325,  the  pole  of  the  earth's  equator  really  has  round  tLe  polo 
of  the  ecliptic,  in  consequence  of  the  joint  eflFects  of  precesiiioD  und  nufei 
tion,  which  are  thus  uranogriphically  represented.  If  we  ?,ip<  r.;,d<:  to  tla 
eflFect  of  lunar  precession  that  of  the  eolar,  which  done  yioM  oav  v.  iL^ 
pole  to  describe  a  circle  uniforialy  about  P,  this  wLU  only  all'ect  the  uudu- 
lations  of  our  waved  curve,  by  extending  thom  in  length,  but  v.  ill  produce 
no  effect  on  the  depth  of  the  waves,  or  the  «)>;*?arsion3  of  the  earth's  axis 
to  and  from  the  pole  of  the  ecliptic.  Thus  we  see  that  the  two  pbonompTir* 
of  nutation  and  precession  are  intimately  coiniect< d,  or  ratlier  both  of 
theui  essential  constituent  parts  of  one  and  the  same  pbenomenijn.  f  ',* 
hiivdiy  necessary  to  state  that  a  rigorous  analysis  of  ihis  great  pr  ^Hlem,  i.y 
an  exact  est)  !u. lion  of  all  the  acting  forces  and  summation  of  their  dy- 
namical effect ,  ;'ada  *o  the  preoJse  value  of  the  co-efficients  of  precessioa 
and  nutation,  whuh  observation  assigns  to  them.  The  solar  and  luiar 
portions  of  the  prt^c.vjsion  of  the  equinoxes,  that  is  to  say,  those  portions 
which  are  liaiform.  are  to  each  other  in  the  proportion  of  ubout  2  to  5. 

(650.)  la  the  nutation  of  the  earth's  axis  we  have  tm  example  (the 
first  of  its  kind  which  has  occurred  to  us),  of  a  periodical  movement  in 
one  part  of  the  system,  giving  rise  to  a  motion  having  the  same  precise 
period  in  niiother.  The  motion  of  the  moon's  nodes  is  Imre,  we  see, 
represented,  though  under  a  very  different  form,  yet  in  the  same  exact 
periodic  time,  by  a  movement  of  a  peculiar  oscillatory  kind  ijnpressed  on 
the  solid  mass  of  the  earth.  We  must  not  let  the  opportunity  pass  of 
generalizing  the  principle  involved  in  this  result,  as  it  is  one  which  we 
shall  find  again  and  again  exemplified  in  every  part  of  physical  astronomy, 
nay,  in  every  department  of  natural  science.  It  may  be  stated  as  "  the 
principle  of  forced  oscillations,  or  of  forced  vibrations,"  and  thus  gene- 
rally announced  : — 

If  one  part  of  any  st/stem  connected  either  hy  material  ties,  or  by  the 
mutual  attractions  of  its  members,  be  continually  maintained  by  any 
cause,  whether  inherent  in  the  constitution  of  the  system  or  external  to  it, 
in  a  state  of  regular  periodic  motion,  that  motion  will  be  propagated 
throughout  the  whole  systems,  and  will  give  rise,  in  every  member  of  it 
and  in  every  part  of  each  member,  to  periodic  movements  executed  in 
equal  period,  with  that  to  which  they  owe  their  origin,  though  not  nec^s- 
warily  synchronous  with  them  in  their  maxima  and  minimal 

'  See  a  demonstration  of  this  theorem  fur  the  forced  vibrations  of  systems  connected 
by  material  ties  of  imperfect  elasticity,  in  my  treatise  on  Sound,  Encyc.  Metrop.  art. 
323.    The  demonstration  is  easily  extended  and  generaUzed  to  take  in  other  systems. 


PRINCIPLE   OF  FORCED  VIBRATIONS. 


853 


The  system  may  be  favourably  or  unfavourably  constituted  for  such  a 
transfer  of  periodic  movements,  or  favourably  in  some  of  its  parts  and 
unfavourably  in  others  j  and  accordingly  as  it  is  the  one  or  the  other,  the 
derivative  oscillation  (as  it  may  be  termed)  will  be  imperceptible  in  one 
case,  of  appreciable  magnitude  in  another,  and  even  more  perceptible  in 
its  visible  effects  than  the  original  cause  in  a  third ;  of  this  last  kind  we  have 
an  instance  in  the  moon's  acceleration,  to  be  hereafter  noticed. 

(651.)  It  so  happens  that  our  situation  on  the  earth,  and  the  delicacy 
which  our  observations  have  att«incd,  enable  us  to  make  it  as  it  were  an 
instrument  to  /eel  these  forced  vibrations,  —  these  derivative  motioos, 
communicated  from  various  quarters,  especially  from  our  near  neighbour, 
the  moon,  much  in  the  same  way  as  we  detect,  by  the  trembling  of  a 
board  beneath  us,  the  secret  transfer  of  motion  by  which  the  sound  of  an 
organ-pipe  is  dispersed  through  the  air,  and  carried  down  into  the  earth. 
Accordingly,  the  monthly  revolution  of  the  moon,  and  the  annual  motion 
of  the  sun,  produce,  each  of  them,  small  nutations  in  the  earth's  axis, 
whose  periods  are  respectively  half  a  month  and  half  a  year,  each  of 
which,  in  this  view  of  the  subject,  is  to  be  regarded  as'  one  portion  of  a 
period  consisting  of  two  equal  and  similar  parts.  But  the  most  remark- 
able instance,  by  far,  of  this  propagation  of  periods,  and  one  of  high 
importance  to  mankind,  is  that  of  the  tides,  which  are  forced  oscillations, 
excited  by  the  rotation  of  the  earth  in  an  ocean  disturbed  from  its  figure 
by  the  varying  attractions  of  the  sun  and  moon,  each  revolving  in  its  own 
orbit,  and  propagating  its  own  period  into  the  joint  phenomenon.  The 
explanation  of  the  tides,  however,  belongs  mora  properly  to  that  part  of 
the  general  subject  of  perturbations  which  treats  of  the  action  of  the 
radial  component  of  the  disturbing  force,  and  is  therefore  postponed  to  a 
subsequent  chapter. 


28 


854 


OUTLINES   OF   ASTRONOMY. 


CHAPTER  XIII. 
THEORY   OP  THE   AXES,   PERIHELIA,   AND  EXCENTRICITIES. 

VARIATION  OP  ELEMENTS  IN  GENERAL. — DISTINCTION  BETWEEN  PE- 
RIODIC AND  SECULAR  VARIATIONS, — GEOMETRICAL  EXPRESSION  OF 
TANGENTIAL  AND  NORMAL  FORCES.  —  VARIATION  OP  THE  MAJOR 
AXIS  PRODUCED  ONLY  BY  "^HE  TANGENTIAL  FORCE. — LAQRANOE'.S 
THEOREM  OP  THE  CONSERVATION  OP  THE  MEAN  DISTANCES  AND 
PERIODS. — THEORY  OP  THE  PERIHELIA  AND  EXCENTRICITIES.— 
GEOMETRICAL  REPRESENTATION  OF  THEIR  MOMENTARY  VARIA- 
TIONS.—  ESTIMATION  OP  THE  DISTURBING  FORCES  IN  NEARLY 
CIRCULAR  ORBITS.  —  APPLICATION  TO  THE  CASE  OP  THE  MOON. — 
THEORY  OP  THE  LUNAR  APSIDES  AND  EXCENTRTCITY.  —  EXPERI- 
MENTAL ILLUSTRATION.  —  APPLICATION  OF  THE  FOREGOING  PRIN- 
CIPLES TO  THE  PLANETARY  THEORY.  —  COMPENSATION  IN  ORBITS 
VERY   NEARLY  CIRCULAR.  —  EFFECTS    OP    ELLIPTICITY.  —  GENERAL 

RESULTS.  —  Lagrange's    theorem   op  the    stability  op  the 

EXCENTRICITIES. 

(652.)  In  the  foregoing  chapter  we  have  sufficiently  explained  'lie  action 
of  the  orthogonal  component  of  the  disturbing  force,  and  traced  it  to  its 
results  in  a  continual  displacement  of  the  plane  of  the  disturbed  orbit,  in 
virtue  of  which  the  nodes  of  that  plane  alternately  advance  and  recede 
upon  the  plane  of  the  disturbing  body's  orbit,  with  a  general  preponde- 
rance on  the  side  of  advance,  so  as  after  the  lapse  of  a  long  period  to 
cause  the  nodes  to  make  a  complete  revolution  an<^  come  round  to  their 
former  situation.  At  the  same  time  the  inclination  of  the  plane  of  the 
disturbed  motion  continually  changes,  alternately  increasing  and  diminish- 
ing; the  increase  and  diminution,  however,  compensating  each  other, 
nearly  in  single  revolutions  of  the  disturbed  and  disturbing  bodies,  more 
exactly  in  many,  and  with  perfect  accuracy  in  long  periods,  such  as  those 
of  a  complete  revolution  of  the  nodes  and  apsides.  In  the  present  and 
following  chapters  we  shall  endeavour  to  trace  the  effects  of  the  other 
components  of  the  disturbing  force,  —  those  which  act  in  the  plane  (for 


-,  ^ 


VARIATION   OF   ELEMENTS. 


355 


the  time  being)  of  the  disturbed  orbit,  and  which  tend  to  derange  the 
elliptic  form  of  the  orbit,  and  the  laws  of  elliptic  motion  in  that  piano. 
The  small  inclination,  generally  speaking,  of  the  orbits  of  the  planets  and 
satellites  to  each  other,  permits  us  to  separate  these  effects  in  theory  one 
from  the  other,  and  thereby  greatly  to  simplify  thoir  consideration.  Ac- 
cordingly, in  what  follows,  we  shall  throughout  '  gleet  the  mutual  incli- 
nation of  the  orbits  of  the  disturbed  and  disturbing  bodies,  and  regard  all 
the  foroog  as  acting  and  all  the  raotions  as  performed  in  one  plane. 

(653.)  In  considering  the  changes  induced  by  the  mutual  action  of  two 
bodies,  in  different  aspects  with  respect  to  each  other,  on  the  magnitudes 
and  forms  of  their  orbits,  and  in  their  positions  therein,  it  will  be  proper 
in  the  first  instance  to  explain  the  conventions  under  which  geometers 
and  astronomers  have  alike  agreed  to  use  the  language  and  laws  of  the 
elliptic  system,  and  to  continue  to  apply  them  to  disturbed  orbits,  although 
those  orbits  so  disturbed  are  no  longer,  in  mathematical  strictness,  ellipses, 
or  any  known  curves.  This  they  do,  partly  on  account  of  the  convenience 
of  conception  and  calculati'^*.  which  attaches  to  this  system,  but  much 
more  for  this  reason,  —  that  it  is  found,  and  may  be  demonstrated  from 
the  dynamical  relations  of  the  case,  that  the  departure  of  each  planet  from 
its  ellipse,  as  determined  at  any  epoch,  is  capable  of  being  truly  repre- 
sented, by  supposing  the  ellipse  itself  to  be  slowly  variable,  to  change  its 
mngnitude  and  excentricity,  and  to  shift  its  position  and  the  plane  in 
which  it  lies  according  to  certain  laws,  while  the  planet  all  the  time  con- 
tinues to  move  in  this  ellipse,  just  as  it  would  do  if  the  ellipse  remained 
invariable  and  the  disturbing  forces  had  no  existence.  By  this  way  of 
considering  the  subject,  the  whole  effect  of  the  disturbing  forces  is  regarded 
as  thrown  upon  the  orbit,  while  the  relations  of  the  planet  to  that  orbit 
remain  unchanged.  This  course  of  procedure,  indeed,  is  the  most  natural, 
and  is  in  some  sort  forced  upon  us  by  the  extreme  slowness  with  which 
the  variation  of  the  elements,  at  least  where  the  planets  only  are  con- 
cerned, develop  themselves.  For  instance,  the  fraction  expressing  the 
excentricity  of  the  earth's  orbit  changes  no  more  than  0.00004  in  its 
amount  in  a  centurij;  and  the  place  of  its  perihelion,  as  referred  to  the 
sphere  of  the  heavens,  by  only  19'  39"  in  the  same  time.  For  several 
years,  therefore,  it  would  be  next  to  impossible  to  distinguish  between  ai; 
ellipse  so  varied  and  one  that  had  not  varied  at  all ;  and  in  a  single  revo- 
lution, the  difference  between  the  original  ellipse  and  the  curve  really 
represented  by  the  varying  one,  is  so  excessively  minute,  that,  if  accU' 
rately  drawn  on  a  table,  six  feet  in  diameter,  the  nicest  examination  with 
microscopes,  continued  along  the  whole  outLncd  of  the  two  curves,  would 
hardly  detect  any  perceptible  interval  between  them.     Not  to  call  a  mo- 


n*;!; 


856 


OUTLINES   or  ASTRONOMY- 


tlon  BO  minutely  conforniiug  itself  to  an  elliptic  curve,  elliptic,  would  bo 
afifectation,  even  granting  the  existence  of  trivial  departures  alternately  on 
one  side  or  on  the  other ;  though  on  the  othor  hand,  to  neglect  a  varia- 
tion, which  continues  to  accumulate  from  ago  to  age,  till  it  forces  itbclf  on 
our  notice,  would  be  wilful  blindness. 

(654.)  Geometers,  then,  have  agreed,  in  each  single  revolution,  or  for 
any  moderate  interval  of  tiuo,  to  regard  the  motion  of  each  planet  as 
elliptic,  and  pcrfo.med  according  to  Kepler's  laws,  with  a  reserve  in 
favour  of  those  very  small  and  transient  fluctuations  which  tuko  place 
within  that  time,  but  at  the  same  time  to  regard  all  the  elements  of  each 
ellipse  as  in  a  continual,  though  extremely  slow,  state  of  change ;  and,  in 
tracing  the  effects  of  perturbation  on  the  system,  they  take  account  prin- 
cipally, or  entirely,  of  this  change  of  the  elementr>,  au  that  upon  which  any 
material  change  in  the  great  features  of  tiie  system  will  ultimately  depend. 

(655.)  And  hore  we  encounter  the  distinction  botweon  what  are  termed 
secular  variations,  and  such  as  are  rapidly  periodic,  and  are  coniponsated 
in  short  intervals.  In  our  exposition  of  the  variation  of  the  inelinatitu 
of  a  disturbed  orbit  (art.  636,)  for  instance,  we  showed  that,  in  each 
single  revolution  of  the  disturbed  body,  the  plane  of  its  motion  underwent 
fluctuations  to  and  fro  in  its  inclination  to  that  of  the  disturbing  body, 
which  nearly  v^ompensated  each  other;  leaving,  however,  a  portion  out- 
standing, which  again  is  nearly  compensated  by  the  revolution  of  the  dis- 
turbing  body,  yet  still  leaving  outstanding  and  uncompensated  a  minute 
portion  of  the  change  which  requij-es  a  whole  revolution  of  the  node  to 
compensate  and  bring  it  back  to  an  average  or  mean  value.  Now,  the 
tv'O  first  compensations  which  are  operated  by  the  planets  going  through 
the  succession  of  configurations  with  each  other,  and  therefore  in  compara- 
tively short  periods,  are  called  periodic  variations ;  and  the  deviations  thus 
compensated  are  called  inequalities  depevding  on  configurations}  while 
the  last,  which  is  operated  by  a  period  of  the  node  (one  of  the  elements,) 
has  nothing  to  do  with  the  configurations  of  the  individual  planets,  requires 
a  very  long  period  of  time  for  its  consummation,  and  is,  therefore,  distin- 
guished from  the  former  by  the  term  secular  variation. 

(656.)  It  is  true,  that,  to  afford  an  exact  representation  of  the  motions 
of  a  disturbed  body,  whether  planet  or  satellite,  both  periodical  and 
secular  variations,  with  their  corresponding  inequalities,  require  to  be 
expressed  j  and,  indeed,  the  former  even  more  than  the  latter ;  seeing  that 
the  secular  inequalities  are,  in  fact,  nothing  but  what  remains  after  the 
mutual  destruction  of  a  much  larger  amount  (as  it  very  often  is)  of  peri- 
odical. Bnt  these  are  in  their  nature  transient  and  temporary:  they 
disappear  in  short  periods,  and  leave  uo  trace.     The  planet  is  temporarily 


GEOMETRICAL  EXPRESRT 


OF  FORCE. 


357 


,  IM 


drawn  from  its  orbit  (its  slowly  varying  orbit,)  but  forthwitb  returns  to 
it,  to  deviiito  presently  as  much  tbe  other  way,  while  the  varied  orbit 
accommodutcs  and  adjusts  itself  to  the  average  of  these  excursions  on 
either  side  of  it;  and  thus  continues  to  present,  for  a  succession  of  indefi- 
nite ages,  a  kind  of  medium  picture  of  all  that  the  planet  has  been  doing 
in  their  lapse,  in  which  the  expression  and  the  character  is  preserved;  but 
the  individual  features  arc  merged  and  lost.  These  periodic  inequalities, 
however,  are,  as  we  have  observed,  by  no  means  neglected,  but  it  is  more 
convenient  to  take  account  of  them  by  a  separate  process,  independent 
of  the  secular  variations  of  the  elements. 

(0137.)  In  order  to  avoid  complication,  while  endeavouring  to  give  the 
reader  an  insight  into  both  kinds  of  variations,  we  shall  '  enceforward 
conceive  all  the  orbits  to  lie  in  one  plane,  and  confine  our  attention  to  the 
case  of  two  only,  that  of  the  disturbed  and  disturbing  body,  a  view  of  the 
subject  which  (as  we  have  seen)  comprehends  the  case  of  the  moon  dis- 
turbed ))y  the  sun,  since  any  one  of  the  bodies  may  be  regarded  as  fixed 
at  pleasure,  provided  we  conceive  all  its  motions  transferred  in  a  contrary 

Fig.  86. 


^i 


Mir ,  J)'!' 


^K 


direction  to  each  of  the  others.  Let  therefore  A  P B  be  the  uivlns! iirbed 
elliptic  orbit  of  a  planet  P;  M  a  disturbing  body,  join  M  P,  and  supposing 
MK=M  S  take  MN  :  M  K  ::  M  K^ :  M  PI  Then  if  SN  be  joined, 
N  S  will  represent  the  disturbing  force  of  M  or  P,  on  the  same  scale  that 
S  M  represents  M's  attraction  on  S.  Suppose  Z  P  Y  a  tangent  at  P,  S  Y 
perpendicular  to  it,  and  N  T,  N  L  perpendicular  respectively  to  S  Y  and 
P  iS  produced.  Then  will  N  T  represent  the  tangential,  T  S  the  normal, 
N  L  the  transversal,  and  L  S  the  radial  components  of  the  disturbing 
force.     In  circular  orbits  or  orbits  only  slightly  elliptic,  the  directions 


858 


OUTLINES   OF  A8TU0N0MY. 


I 


PS  L  and  SY  are  lu-arly  coincident,  and  the  fi)rnicr  pair  of  ioiccs  will 
diflcr  but  slightly  from  the  latter.  Wo  bIuiU  here,  however,  take  tlio 
geiu'ral  case,  and  proceed  to  investigate  in  an  elliptic  orbit  of  any  di'g"'>c 
of  excontricity  the  momentary  changes  produced  by  the  action  of  the  dis. 
turbiiig  force  in  those  elements  on  which  the  magnitude,  situation,  and 
form  of  the  orbit  depend  (t.  c.  the  length  and  position  of  the  major  axis 
and  the  excontricity, )  in  the  same  way  as  in  the  last  chapter  we  diitcr- 
mined  the  momentary  changes  of  the  inclinutiou  and  node  eimilarly  pro- 
duced by  the  orthogonal  force. 

(0o8.)  Wc  shall  begin  with  the  momentary  variation  in  tho  knyth  of 
the  axis,  an  eletnont  of  the  first  importance,  as  on  it  depends  (art.  4H7) 
the  periodic  time  and  mean  angular  motion  of  tho  planot,  as  well  as  the 
average  supply  of  light  and  heat  it  receives  in  a  given  time  from  the  sun, 
any  permanent  or  constantly  progressive  change  in  which  would  alter 
most  materially  the  conditions  of  existence  of  living  beings  on  its  surface. 
Now  it  is  a  property  of  elliptic  motion  performed  under  tho  influence  of 
gravity,  and  in  conformity  with  Kepler's  laws,  that  if  tho  velocity  with 
which  a  planet  moves  at  any  point  of  its  orbit  bo  given,  and  also  the 
distance  of  that  point  from  tho  sun,  tho  major  axis  of  the  orbit  is  thereby 
also  given.  It  is  no  matter  in  what  direction  the  planet  may  bo  moving 
at  that  moment.  This  will  influence  the  excontricity  and  tho  position  of 
its  ellipse,  but  not  its  length.  This  property  of  elliptic  motion  has  been 
demonstrated  by  Newton,  and  is  one  of  the  most  obvious  and  elementary 
conclusions  from  his  theory.  Let  us  now  consider  a  planet  describing  an 
indefinitely  small  arc  of  its  orbit  about  tho  sun,  under  tho  joint  influence 
of  its  attraction,  and  the  disturbing  power  of  another  planot.  This  arc 
will  have  some  certain  curvature  and  direction,  and,  therefore,  may  be 
considered  as  an  arc  of  a  certain  ellipse  described  about  tho  sun  as  a 
focus,  for  this  plain  reason, — that  whatever  be  the  curvature  and  direction 
of  the  arc  in  question,  an  ellipse  may  always  be  assigned,  whose  focus 
shall  be  in  the  sun,  and  which  shall  coincide  with  it  throughout  the  whole 
interval  (supposed  indefinitely  small)  between  its  extreme  points.  This 
is  a  matter  of  pure  g<!ometry.  It  does  not  follow,  however,  that  the 
ellipse  thus  instantaneously  determined  will  have  the  same  elements  as 
that  similarly  determinei  fnm  the  arc  described  in  either  the  previous  or 
the  subsequent  instant.  If  the  disturbing  force  did  not  exist,  this  would 
be  the  case  :  but,  by  its  action,  a  variation  of  the  element  from  instant  to 
instant  is  produced,  and  the  ellipse  so  determined  is  in  a  continual  state 
of  change.  Now  when  tho  planet  has  reached  the  end  of  the  small  arc 
under  consideration,  the  question  whether  it  will  in  the  next  instant 
describe  an  arc  of  an  ellipse  having  tho  same  or  a  varied  axis  will  depend, 


VARIATION   OF   THE   MAJOIi   AXIS. 


8r.o 


not  on  tlie  new  dlrectton  impreHHcd  upon  it.  by  tlio  acting  forces, — for  the 
iixifl,  ofl  wc  have  scon;  is  iudcptmdoul  uf  that  direction, — not  on  its  change 
of  diHtiiuco  from  the  sun,  while  dcHcribing  the  former  arc,  —  for  the  ele- 
ments of  that  arc  aro  accommodated  to  it,  so  that  one  and  the  same  axis 
uiu8t  belong  to  its  beginning  and  its  end.  The  question,  in  short,  whether 
iu  the  next  aro  it  shall  take  up  a  new  major  axis  or  go  on  with  the  old 
one  will  depend  solely  on  this — whether  its  velociti/  has  or  has  not  under- 
gone a  change  by  the  action  of  tho  di»turbing  force.  For  the  central 
force  residing  in  tho  focus  can  impress  on  it  no  such  change  of  velocity 
as  to  be  incompatible  with  tho  permanence  of  its  ellipse,  seeing  that  it  is 
by  the  action  of  that  force  that  the  velocity  is  maintained  in  that  due 
proportion  to  the  distance  which  elliptic  motion,  as  such,  requires. 

(059.)  Thus  we  see  that  tho  momentary  variation  of  the  major  axis 
depends  on  nothing  but  the  momentary  deviation  from  the  law  of  elliptic 
velocity  produced  by  tho  disturbing  force,  without  tho  least  regard  to  tho 
direction  in  which  that  extraneous  velocity  is  impressed,  or  tho  dintance 
from  the  sun  at  which  the  planet  may  be  situated,  at  the  moment  of  its 
impression.  Nay,  we  may  even  go  farther,  for,  as  this  holds  good  at  every 
instant  of  its  motion,  it  will  follow  that  after  the  lapse  of  any  time,  how- 
ever great,  the  total  amount  of  change  which  the  axis  may  have  under- 
gone will  bo  determined  only  by  the  total  deviation  produced  by  the  action 
of  the  disturbing  force  in  the  velocity  of  the  disturbed  body  from  that 
which  it  would  have  had  in  its  undisturbed  ellipse,  at  the  same  distance 
from  the  centre,  and  that  therefore  the  total  amount  of  change  produced 
in  the  axis  in  any  lapse  of  time  may  be  estimated,  if  we  know  at  every 
instant  the  efficacy  of  the  disturbing  force  to  alter  the  velocity  of  the 
body's  motion,  and  that  without  any  regard  to  the  alterations  which  the 
action  of  that  force  may  have  produced  in  the  other  elements  of  the 
motion  in  the  same  time. 

(660.)  Now  it  is  not  the  whole  disturbing  force  which  is  effective  in 
changing  P's  velocity,  but  only  its  tangential  component.  The  normal 
component  tends  merely  to  alter  the  curvature  of  the  orbit  or  to  deflect  it 
into  conformity  with  a  circle  of  curvature  of  greater  or  lesser  radius,  as 
the  case  may  be,  and  in  no  way  to  alter  the  velocity.  Hence  it  appears 
that  the  variation  of  the  length  of  the  axin  is  due  entirely  to  the  tangen- 
tial  force,  and  is  quite  independent  on  the  normal.  Now  it  is  easily  shown 
that  as  the  velocity  increases,  the  axis  increases  (the  distance  remaining 
unaltered')  though  not  in  the  same  exact  proportion.     Hence  it  follows 

'  If  a  be  the  semiaxis,  r  the  radius  vector,  and  v  the  velocity  of  P  in  any  point  of  an 
ellipse,  a  is  given  by  the  relation  t)'= ,  the  units  of  velocity  and  Jorce  being  pro- 
perly assumed. 


Iiti'= 


m^is. 


{ 

!■    ^ 

^i' 

>'^ 

1 

' 

?    c 

1 

4 

!1 

Wh 

P'l 

^    ' 

!t 

I'M  It' 'i 


860 


OUTLINES   OF  ASTRONOMY. 


3    ' 


that  if  the  tangential  disturbing  force  conspires  with  the  motion  of  P,  its 
momentary  action  increases  the  axis  of  the  disturbed  orbit,  whatever  be 
the  situation  of  P  in  its  orbit,  and  vice  versd. 

(661.)  Let  A  S  B  (fig.  art.  657)  be  the  major  axis  of  the  ellipse  A 1* 
B,  and  on  the  opposite  side  of  A  B  take  two  points  P'  and  M',  similarly 
situated  with  respect  to  the  axis  with  P  and  M  on  their  side.  Then  if  at 
F  and  M'  bodies  equal  to  P  and  M  be  placed,  the  forces  exerted  by  M' 
on  P'  and  S  will  be  equal  to  those  exerted  by  M  on  P  and  S,  and  there- 
fore the  tangential  disturbing  force  of  M'  on  P'  exerted  in  the  directiou 
P'  Z'  (suppose)  will  equal  that  exerted  by  M  on  P  in  the  direction  P  Z. 
P'  therefore  (supposing  it  to  revolve  in  the  same  direction  round  S  as  P) 
will  be  retarded  (or  accelerated,  as  the  case  may  be)  by  precisely  the  same 
force  by  which  P  is  accelerated  (or  retarded),  so  that  the  variation  in  the 
axis  of  the  respective  orbits  of  P  and  P'  will  be  equal  in  amount,  but  con- 
trary in  character.  Suppose  now  M's  orbit  to  be  circular.  Then  (if  the 
periodic  times'iqf  M  and  P  be  not  commensurate,  so  that  a  moderate 
number  of  revolutions  may  bring  them  bach  to  the  same  precise  relative 
positions)  it  will  necessarily  happen,  that  m  the  course  of  a  very  great 
number  of  revolutions  of  both  bodies,  P  will  have  been  presented  to  M 
on  one  side  of  the  axis,  at  some  one  moment,  in  the  same  manner  as  at 
some  other  moment  on  the  other.  Whatever  variation  may  have  been 
eflFected  in  its  axis  in  the  one  situation  will  have  been  reversed  in  that 
symmetrically  opposite,  and  the  ultimate  result,  on  a  general  average  of 
an  infinite  number  of  revolutions,  will  be  a  complete  and  exact  compen- 
sation of  the  variations  in  one  direction  by  those  in  the  direction  opposite. 

(662.)  Suppose,  next,  P's  orbit  to  be  circular.  If  now  M's  orbit  were 
60  also,  it  is  evident  that  in  one  complete  synodic  revolution,  an  exact 
restoration  of  the  axis  to  its  original  length  would  take  place,  because  the 
tangential  forces  would  be  symmetrically  equal  and  opposite  during  each 
alternate  quarter  revolution.  But  let  M,  during  a  synodic  revolution, 
have  receded  somewhat  from  S,  then  will  its  disturbing  power  have  become 
gradually  weaker,  so  that,  in  a  synodic  revolution,  the  tangential  force  in 
each  quadrant,  though  reversed  in  direction  being  inferior  in  power,  an 
exact  compensation  will  not  have  been  effected,  but  there  will  be  left  an 
outstanding  uncompensated  portion,  the  excess  of  the  stronger  over  the 
feebler  eff'ects.  But  now  suppose  IM  to  approach  by  the  same  gradations 
as  it  before  receded.  It  is  clear  that  this  result  will  be  reversed  j  since 
fje  uncompensated  stronger  actions  will  all  lie  in  the  opposite  direction. 
Now  suppose  M's  orbit  to  be  elliptic.  Then  during  its  recess  from  S,  or 
in  the  half  revolution  from  its  perihelion  to  its  aphelion,  a  continual  un- 
compensated variation  will  go  on  accumulating  in  one  direction.     But, 


'ir 


VARIATION  OF  THE  MAJOR  AXIS. 


361 


from  what  has  been  said,  it  is  clear  that  this  will  be  destroyed,  during  M's 
approach  to  S  in  the  other  half  of  its  orbit,  so  that  here  again,  on  the 
average  of  a  multitude  of  revolutions  during  which  P  has  been  presented 
to  M  in  every  situation  for  every  distance  of  M  from  S,  the  restoration 
will  be  effected. 

(GG3.)  If  neither  P's  nor  M's  orbit  be  circular,  and  if  moreover  the 
directions  of  their  axes  be  different,  this  reasoning,  drawn  from  the  sym- 
metry of  their  relations  to  each  other,  does  not  apply,  and  it  becomes 
necessary  to  take  a  more  general  view  of  the  matter.  Among  the  funda- 
mental relations  of  dynamics,  relations  which  presuppose  no  particular 
law  of  force  like  that  of  gravitation,  but  which  express  in  general  terms 
the  results  of  the  action  of  force  on  matter  during  time,  to  produce  or 
change  velocity,  is  one  usually  cited  as  the  "  Principle  of  the  conserva- 
tion of  the  vis  viva,"  which  applies  directly  to  the  case  before  us.  This 
principle  (or  rather  this  theorem)  declares  that  if  a  body  subjected  at 
every  instant  of  its  motion  to  the  action  of  forces  directed  to  fixed  centres 
(no  matter  how  numerous),  and  having  their  intensity  dependent  only  on 
the  distances  from  their  respective  centres  of  action,  travel  from  one  point 
of  space  to  another,  the  velocity  which  it  has  on  its  arrival  at  the  latter 
point  will  differ  from  that  which  it  had  on  setting  out  from  the  former,  by 
a  quantity  depending  only  on  the  different  relative  situations  of  these  two 
points  in  space,  without  the  least  reference  to  the  form  of  the  curve  in 
which  it  may  have  moved  in  passing  from  one  point  to  the  other,  whether 
that  curve  have  been  described  freely  under  the  simple  influence  of  the 
central  forces,  or  the  body  have  been  compolled  to  glide  upon  it,  as  a  bead 
upon  a  smooth  wire.  Among  the  forces  thus  acting  may  be  included  any 
constant  forces,  acting  in  parallel  directions,  which  may  be  regarded  as 
directed  to  fixed  centres  infinitely  distant.  It  follows  from  this  theorem, 
that,  if  the  body  return  to  the  point  P,  from  which  it  set  out,  its  velocity 
of  arrival  will  be  the  same  with  that  of  its  departure ;  a  conclusion  which 
(for  the  purpose  we  have  in  view)  sets  us  free  from  the  necessity  of  enter- 
ing into  any  consideration  of  the  laws  of  the  disturbing  force,  the  change 
which  its  action  may  have  induced  in  the  form  of  the  orbit  of  P,  or  the 
successive  steps  by  which  velocity  generated  at  one  point  of  its  interme- 
diate path  is  destroyed  at  another,  by  the  reversed  action  of  the  tangen- 
tial force.  Now  to  apply  this  theorem  to  the  case  in  question,  let  M  be 
supposed  to  retain  a  fixed  position  during  one  whole  revolution  of  P. 
P  then  is  acted  on,  during  that  revolution,  by  three  forces:  l.st.  by  the 
central  attraction  of  S  directed  always  to  S ;  2nd.  by  that  to  M,  always 
directed  to  M ;  3rd.  by  a  force  equal  to  M's  attraction  on  S ;  but  in  the 
direction  M  S,  which  therefore  is  a  constant  force,  acting  always  in  parallel 


1    I* 


fif'll 


h  '    'ft 


862 


OUTLINES    OP   ASTRONOMY. 


directions.  On  completing  its  revolution,  then,  P's  velocity,  and  therefore 
the  major  axis  of  its  orbit,  will  be  found  unaltered,  at  least  neglecting 
that  excessively  mimite  difiFerence  which  will  result  from  the  non-arrival 
after  a  revolution  at  the  exact  point  of  its  departure  by  reason  of  the  per- 
turbations in  the  orbit  produced  in  the  interim  by  the  disturbing  force, 
which  for  the  present  we  may  neglect. 

(664.)  Now  suppose  M  to  revolve,  and  it  will  appear,  by  a  reasoning 
precisely  similar  to  that  of  art,  662,  that  whatever  uncompensated  varia- 
tion of  the  velocity  arises  in  successive  revolutions  of  P  during  M's  recess 
from  S  will  be  destroyed  by  contrary  uncompensated  variations  arising 
during  its  approach.  Or,  more  simply  and  generally  thus  :  whatever  M's 
situation  may  be,  for  every  place  which  P  can  have,  there  must  exist  some 
other  place  of  P  (as  P'),  in  which  the  action  of  M  shall  be  precisely 
reversed.  Now  if  the  periods  he  incommensurahle,  in  an  indefinite 
number  of  revolutions  of  both  bodies,  for  every  possible  combination  of 
situations  (M,  P)  there  will  occur,  at  some  time  or  other,  the  combination 
(M,  P')  which  neutralizes  the  effect  of  the  other,  when  carried  to  die 
general  account ;  so  that  ultimately,  and  when  very  long  periods  of  time 
are  embraced,  a  complete  compensation  will  be  found  to  be  worked  out. 

(665.)  This  supposes,  however,  that  in  such  long  periods  the  orbit  of 
M  is  not  so  altered  as  to  render  the  occurrence  of  the  compensating  situ- 
ation (M,  P')  impossible.  This  would  be  the  case  if  M's  orbit  were  to 
dilate  or  contract  indefinitely  by  a  variation  in  its  axis.  But  the  same 
reasoning  which  applies  to  P,  applies  also  to  ]M.  P  retaining  a  fxed 
situation,  M's  velocity,  and  therefore  the  axis  of  its  orbit,  would  be  ex- 
actly restored  at  the  end  of  a  revolution  of  M ;  so  that  for  every  position 
P  M  there  exists  a  compensating  position  P  M'.  Thus  M's  orbit  is  main- 
tained of  the  same  magnitude,  and  the  possibility  of  the  occurrence  of 
the  compensating  situation  (M,  P')  is  secured. 

(660.)  To  demonstrate  as  a  rigorous  mathematical  truth  the  complete 
and  absolute  ultimate  compensation  of  the  variations  in  question,  it  would 
be  requisite  to  show  that  the  minute  outstanding  changes  due  to  the  non- 
arrivals  of  P  and  M  at  the  same  exact  points  at  the  end  of  each  revolu- 
tion, cannot  accumulate  in  the  course  of  infinite  ages  in  one  direction. 
Now  it  will  appear  in  the  subsequent  part  of  this  chapter,  that  the  effect 
of  perturbation  on  the  excentricities  and  apsides  of  the  orbits  is  to  cause 
the  former  to  undergo  only  periodical  variations,  and  the  latter  to  revolve 
and  take  up  in  succession  every  possible  situation.  Hence  in  the  course 
of  infinite  ages,  the  points  of  arrival  c^  P  and  M  at  fixed  lines  of  direc- 
tion, S  P,  8  M,  in  successive  revolution*^,  though  at  one  time  they  will 
approach  S;  at  another  will  recede  from  it,  fluctuating  to  and  fro  about 


are. 


VARIATION   OF  THE   MAJOR  AXIS. 


863 


mean  points  from  which  they  never  greatly  depart.  And  if  the  arrival 
of  either  of  them  at  P,  at  a  point  nearer  S,  at  the  end  of  a  complete 
revolution,  cause  an  excess  of  velocity,  its  arrival  at  a  more  distant  point 
will  cause  a  deficiency,  and  thus,  as  the  fluctuations  of  distance  to  and  fro 
ultimately  balance  each  other,  so  will  also  the  excesses  and  defects  of 
velocity,  though  in  periods  of  enormous  length,  being  no  less  than  that 
of  a  complete  revolution  of  P's  apsides  for  the  one  cause  of  inequality, 
and  of  a  complete  restoration  of  its  excentricity  for  the  other. 

(667.)  The  dynamical  proposition  on  which  this  reasoning  is  based  is 
general,  and  applies  equally  well  to  cases  wherein  the  forces  act  in  onQ 
plane,  or  are  directed  to  centres  anywhere  situated  in  space.  Hence,  if 
we  take  intc  consideration  the  inclination  of  P's  orbit  to  that  of  M,  the 
same  reasoning  will  apply.  Only  that  in  this  case,  upon  a  complete  revo- 
lution of  P,  the  variation  of  inclination  and  the  motion  of  the  nodes  of 
Fs  orbit  will  prevent  its  returning  to  a  point  in  the  exact  plane  of  its 
original  orbit,  as  that  of  the  excentricity  and  perihelion  prevent  its  arrival 
at  the  same  exact  distance  from  S.  But  since  it  has  been  shown  that  the 
inclination  fluctuates  round  a  mean  state  from  which  it  never  departs 
much,  and  since  the  node  revolves  and  makes  a  complete  circuit,  it  is 
obvious  that  in  a  complete  period  of  the  latter  the  points  of  arrival  of  P 
at  the  same  longitude  will  deviate  as  often  and  by  the  same  quantities 
above  as  below  its  original  point  of  departure  from  exact  coincidence; 
and,  therefore,  that  on  the  average  of  an  infinite  number  of  revolutions, 
the  effect  of  this  cause  of  non-compensation  will  also  be  destroyed. 

(668.)  It  is  evident,  also,  that  the  dynamical  proposil!' r<  in  question 
being  general,  and  applying  equally  to  any  vumber  of  fixed  centres,  as 
well  as  to  anj  distribution  c^  them  in  space,  the  conclusion  would  be  pre- 
cisely the  same  whatever  be  the  number  of  disturbing  bodies,  only  that 
the  periods  of  compensation  would  become  moi'  i  itricately  involved. 
We  are,  therefore,  conducted  to  this  most  remarkable  and  important  con- 
clusion, viz.  that  the  major  axes  of  the  planetary  (and  lunar)  orbits,  and, 
consequently,  also  their  mean  motions  and  periodic  times,  are  subject  to 
none  but  periodical  changes;  that  the  length  of  the  year,  for  example,  in 
the  lapse  of  infinite  ages,  has  no  preponderating  tendency  either  to  increase 
or  diminution, — that  the  planets  will  neither  recede  idcfinitely  from  the 
sun,  nor  fall  into  it,  but  continue,  so  far  as  their  mutual  perturbations  at 
least  are  concerned,  to  levolve  for  ever  in  orbits  of  very  nearly  the  samo 
dimensions  as  at  present. 

^j69.)  This  theorem  (the  Magna  Cliarta  of  our  system),  the  discovery 
of  which  is  due  to  Lagrange,  is  justly  regarded  as  the  most  important,  as 
a  single  result,  of  any  which  have  hitherto  rewarded  the  researches  of 


1 

1 

if 

II''- 

Ir 

. 

::N 

If::;-^ 

1'-  * 

■,    "     !  ri. 

;'■-^ 

■■,-'1  ;■ 

■  f   * 

LaJ^- 

*;■!  f, 

ll 


^) 


i 


«r' 


\4m 


864 


OUTLINES   OF  ASTRONOMY. 


I'i! 


mathematicians  in  this  application  of  their  science ;  and  it  is  especially 
worthy  of  remark,  and  follows  evidently  from  the  view  here  taken  of  it 
that  it  would  not  be  true  but  for  the  influence  of  the  perturbing  forces  on 
other  elements  of  the  orbit,  viz.  the  perihelion  and  excentricity,  and  tie 
inclination  and  nodes ;  since  we  have  seen  that  the  revolution  of  the  ap- 
sides and  nodes,  and  the  periodical  increase  and  diminution  of  the  ex- 
centricities  and  inclinations,  are  both  essential  towards  operating  that  final 
and  complete  compensation  which  gives  if.  a  character  of  mathematical 
exactness.  We  have  here  an  instance  of  a  perturbation  of  one  kind 
operating  on  a  perturbation  of  another  to  annihilate  an  efi'cct  which  would 
otherwise  accumulate  to  the  destruction  of  the  system.  It  must,  however, 
be  borne  in  mind,  that  it  is  the  smallness  of  the  excentricities  of  the  more 
influential  planets,  which  gives  this  theorem  its  practical  importance,  and 
distinguishes  it  from  a  mere  barren  speculative  result.  Within  the  limits 
of  ultimate  restoration,  it  is  this  alone  which  keeps  the  periodical  fluctua- 
tions of  the  axis  to  and  fro  about  a  mean  value  within  moderate  and 
reasonable  limits.  Although  the  earth  might  not  fall  into  the  sun,  or  re- 
cede from  it  beyond  the  present  limits  of  our  system,  any  considerable 
increase  or  diminution  of  its  mean  distance,  to  the  extent,  for  instance,  of 
a  tenth  of  its  actual  amount,  would  not  fail  to  subvert  the  conditions  on 
which  the  existence  of  the  present  race  of  animated  beings  depends. 
Constituted  as  our  system  is,  however,  changes  to  anything  like  this  ex- 
tent are  utterly  precluded.  The  greatest  departure  from  the  mean  value 
of  the  axis  of  any  planetary  orbit  yet  recognized  by  theory  or  observation 
(that  of  the  orbit  of  Saturn  distu/bi'd  by  Jupiter),  does  not  amount  to  a 
thousandth  part  of  its  length  '  The  efi'ects  of  these  fluctuations,  how- 
ever, are  very  sensible,  and  manifest  themselves  in  alternate  accelerations 
an'^  retardations  in  the  angular  motions  of  the  disturbed  about  the  central 
body,  which  cause  it  akernately  to  outrun  and  to  lag  behind  its  elliptic 
place  in  its  orbit,  giving  rise  to  what  are  called  equations  in  its  motion, 
some  of  the  chief  instances  of  which  will  be  hereafter  specified  when  we 
come  to  trace  more  particularly  in  detail  the  eff«}flts  of  the  tangential  force 
in  various  configurations  of  rhe  disturbed  and  d.-iturbing  bodies,  and  t) 
explain  the  consequences  of  a  near  approach  to  commensurability  in  their 
periodic  f  mes.  An  exact  eomroensurability  in  this  respect,  such,  for  in- 
stance, a-  would  bring  both  piunets  round  to  the  same  configuration  in  two 
or  three  revolutions  of  one  of  them,  would  appear  *t  first  sight  to  destroy 
one  of  the  essential  elements  of  our  demoDHtratiou.     But  eve>'  *'ipposing 

■•  Greater  deviations  will  probably  b*  found  to  exint  in  the  orbito  of  the  <mall  extra- 
tropical  planets.  But  these  "re  too  insignificant  memheia  oi</>xi  ityatem  U^  v,<.t  special 
notice  in  a  work  of  this  nature. 


DISPLACEMENT  OF   THE   UPPER  FOCUS. 


865 


vation 

lint  to  a 

how- 

erations 

central 

lUptk 

motion, 

en  we 

I  force 
and  t:) 

their 
for  in- 
in  two 
destroy 
iposing 

II  extra- 
■ipecial 


such  an  exact  adjustment  to  subsist  at  any  epoch,  it  could  not  remain  per- 
manent, sincfci  by  a  remarkable  p.-operty  of  perturbations  of  this  class, 
which  geometers  have  demonstrated,  but  the  reasons  of  which  we  cannot 
stop  to  explain,  any  change  produced  on  the  axis  of  the  disturbed  pluuet's 
orbit  is  necessarily  accompanied  by  a  change  in  the  contrary  dircdion  in 
that  of  the  disturbing,  so  that  the  periods  would  recede  from  commensu- 
rability  by  the  mere  effect  of  their  mutual  action.  Cases  ate  i.ot  wanting 
in  the  planetary  system  of  a  certain  approach  to  commensurability,  and  in 
one  very  remarkable  case  (that  of  Uranus  and  Neptune)  of  a  considerably 
near  one,  not  near  enough,  however,  in  the  smallest  degree  to  affect  the 
validity  of  the  argument,  but  only  to  give  rise  to  inequalities  of  very  long 
periods,  of  which  more  presently.' 

(670.)  The  variation  of  the  length  of  the  axis  of  the  disturbed  orbit 
is  due  solely  to  the  action  of  the  tangential  disturbing  force.  It  is  other- 
wise with  that  of  its  excentricity  and  of  the  position  of  its  axis,  or,  which 
is  the  same  thing,  the  longitude  of  its  perihelion.  Both  the  normal  and 
tangential  components  of  the  disturbing  force  affect  these  elements.  Wo 
shall,  however,  consider  separately  the  influence  of  each,  and,  commencing, 

Fig.  87. 


as  the  simplest  case,  with  that  of  the  tangential  force; — let  P  be  the 
place  of  the  disturbed  planet  in  its  elliptic  orbit  A  P  B,  whose  axis  at  the 
moment  is  A  S  B  and  focus  S.  Suppose  Y  P  Z  to  be  a  tangent  to  this 
wbit  at  P.  Then,  if  we  suppose  A  B  =  2  a,  the  other  focus  of  the 
ellipse,  H,  will  be  found  by  making  the  angle  ZPH  =  YPSorYPII 
=  180°  —  Y  P  Z,  or  S  P  H  =  180°  —  2  Y  P  S,  and  taking  P  H  =  2  a 
—  S  P.  This  is  evident  from  the  nature  of  the  ellipse,  in  which  lines 
drawn  from  any  point  to  the  two  foci  make  equal  angles  with  the  tangent, 

'  41  revolutions  of  Neptune  are  nearly  equal  to  81  of  Uranus,  giving  rise  to  an  ine- 
quality, having  6805  years  for  its  period. 


I    .  1 


it  '.'  \ 


n    i' 

I;  \ 


7='     t      J? 


I 


n 


;fP4l 


866 


OUTLINES   OP  ASTRONOMY. 


and  have  their  sum  equal  to  the  major  axis.  Suppose,  now,  the  tangen- 
tial force  to  act  on  P  and  to  increase  its  velocity.  It  will  therefore  increase 
the  axis,  so  that  the  new  value  assumed  by  a  (viz.  a')  will  be  greater  than 
a.  But  the  tangential  force  does  not  alter  the  angle  of  tangency,  so  that 
to  find  the  new  position  (H')  of  the  upper  focus,  we  must  measure  off 
along  the  same  line  P  H,  a  distance  P  H'  (=  2  a,  —  S  P)  greater  than 
PH.  Do  this  then,  and  join  S  H'  and  produce  it.  Then  will  A'  B'  be 
the  new  position  of  the  axis,  and  i  S  H'  the  new  excentricity.  Hence  we 
conclude,  1st,  that  the  new  position  of  the  perihelion  A'  will  deviate  from 
the  old  one  A  towards  the  same  side  of  the  axis  A  B  on  which  P  is  when 
the  tangential  force  acts  to  increase  the  velocity,  v '.'.ether  P  be  moving 
from  perihelion  to  aphelion,  or  the  contrary.  2dly,  That  on  the  same  sup- 
position as  to  the  action  of  the  tangential  force,  the  excentricity  increases 
\7hen  P  is  between  the  perihelion  and  the  perpendicular  to  the  axis  F  H  G 
drawn  through  the  upper  focus,  and  diminishes  when  between  tlie  aphelion 
and  the  same  perpendicular.  3dly,  That  for  a  given  change  of  velocitj', 
;'.  e.  for  a  given  value  of  the  tangential  force,  the  momentary  variatioi  in 
the  place  of  the  perihelion  is  a  maximum  when  P  is  at  F  or  G,  from 
which  situation  of  P  to  the  perihelion  or  aphelion,  it  decreases  to  nothing, 
the  perihelion  being  stationary  when  P  is  at  A  or  B.  4thly,  That  the 
variation  of  the  excentricity  due  to  this  cause  is  complementary  in  its  luw 
of  increase  and  decrease  to  that  of  the  perihelion,  being  a  maximum  for  a 
given  tangential  force  when  P  is  at  A  or  B,  and  vanishing  when  at  G  or 
F.  And  lastly,  that  where  the  tangential  force  acts  to  diminish  the  velo- 
city, all  these  results  are  reversed.  If  the  orbit  be  very  nearly  circular' 
the  points  F,  G,  will  be  so  situated  that,  -ilthough  not  at  opposite  extrenr. 
ties  of  a  diameter,  the  times  of  describing  A  F,  F  B,  B  G,  and  G  A  will 
be  all  equal,  and  each  of  course  one  quarter  of  the  whole  periodic  timo 
of  P. 

(G71.)  Let  us  now  consider  the  effects?  of  the  normal  component  of  tho 
disturbing  force  upon  thf  same  elements.  The  direct  effect  of  this  force 
is  to  increase  or  diminit;  the  curvature  of  the  orbit  at  the  point  P  of  its 
action,  without  producing  any  change  on  .iC  velocity,  so  that  the  length 
of  the  axis  remains  unaltered  by  its  action.  Now,  an  increase  of  lurva- 
ture  at  P  is  synonymous  with  a  decrease  in  the  angle  of  tangency  S  P  T 
when  P  ia  approaching  towards  S,  and  with  an  increase  in  that  angle 
when  receding  from  S.  Suppose  the  former  case,  and  while  P  approaches 
S  (or  is  moving  from  aphelion  to  perihelion),  let  the  normal  force  act 
'Inwards  or  towards  the  concavity  of  the  ellipse.  Then  will  the  tangoi'' 
P  Y  by  the  action  of  that  forci  have  tiiken  up  the  position  P  Y'.   To  finu 

'  So  nearly  that  the  cube  of  the  excentricity  may  bo  neglected. 


^.: 


'■nt  of  tho 

lis  force 

P  if  its 

le  ler.ith 

if  lurva- 

lat  angle 

(proaclies 

force  act 

tangof 

To  find 


DISPLACEMENT  OF   THE   UPPER  FOCUS. 
Fig.  88. 


367 


the  corresponding  position  H'  taken  up  by  the  focus  of  the  orbit  so  dis- 
turbed, we  raust  make  the  angle  SPH'=180°— 2  SPY',  or,  which 
comes  to  the  same,  draw  P  H'  on  the  side  of  P  H  opposite  to  S,  making 
the  angle  H  P  H'=twico  the  angle  of  deflection  Y  P  Y'  and  in  P  IF  take 
P  H'  =  P  H.  Joining,  then,  S  H'  and  producing  it.  A'  S  IP  M'  will  bo 
the  new  position  of  the  axis,  A'  the  new  perihelion,  and  3  S  H'  the  new 
cxcentricity.  Hence  we  conclude,  1st,  that  the  normal  force  acting 
inwards,  and  P  moving  towards  the  perihelion,  the  new  direction  S  A' 
of  the  perihelion  is  in  advance  (with  reference  to  the  direction  of  P'a 
revolution)  of  the  old  —  or  the  apsides  advance  —  when  P  is  anywhero 
situated  between  F  and  A  (since  when  at  F  the  point  II'  falls  upon  II  M 
between  H  and  M.)  "When  P  is  at  F  the  apsides  are  stationary,  but 
when  P  is  anywhere  between  M  and  F  tho  apsides  retrograde,  H'  in  this 
case  lying  on  the  opposite  sid*^  of  the  axis.  2dly,  That  the  same  direc- 
tions of  the  normal  force  and  of  P's  motic:^  '^^i^x^  supposed,  tho  cxcentri- 
city increases  while  P  moves  through  the  whole  semiellipse  from  aphelion 
to  perihelion — the  rate  of  its  increase  being  a  maximum  when  P  is  at  F, 
and  nothing  at  the  aphelion  and  perihelion.  3dly,  That  these  effects  are 
reversed  in  the  opposite  hali*  of  the  orbit,  A  G  M,  in  which  P  passes  from 
perihelion  to  aphelion  or  recedes  from  S.  4thly,  That  they  are  also 
reversed  by  a  reversal  of  the  direction  of  the  normal  force,  outwards,  in 
place  of  inwards.  5thly,  That  hero  also  the  variations  of  the  excentricity 
and  perihelion  are  compleme^t^^ry  to  each  other;  the  one  variation  being 
most  rapid  when  the  other  vanishes,  and  vice  versa.  6thly,  And  lastly, 
that  the  changes  in  the  situation  of  the  focus  If  produced  by  the  actions 
of  the  tangential  and  normal  components  of  the  disturbing  force  are  at 
right  angles  10  each  other  in  every  situation  of  P,  and  therefore  where 
the  tangential  force  i^'  most  eflicacious  (in  proportion  to  its  intensity)  in 
varying  either  the  one  or  the  other  of  the  elements  in  question,  the 
normal  is  least  30.  and  vice  versd. 


B. 


■;,    j 


368 


OUTLINES   OF  ASTRONOMT. 


(G72.)  To  determine  the  momentary  effect  of  the  whole  disturbing 
force,  then,  we  have  only  to  resolve  it  into  its  tangential  and  normal 
componenta,  and  estimating  by  these  principles  separately  the  effects  of 
either  constituent  on  both  elements,  add  or  substract  the  results  according 
as  they  conspire  or  oppose  each  other.  Or  we  may  at  once  make  the 
angle  11  P  H"  equal  to  twice  the  angle  of  deflection  produced  by  the 
normal  force,  and  lay  off  P  H"=P  H  + twice  the  variation  of  a  produced 
in  the  same  moment  of  time  by  the  tangential  force,  and  H"  will  be  the 
new  focus.  The  momentary  velocity  generated  by  the  tangential  force  is 
calculable  from  a  knowledge  of  that  force  by  the  ordinary  principles  of 
dynamics;  and  from  this,  the  variation  of  the  axis  is  easily  derived.' 
The  momentary  velocity  generated  by  the  normal  force  in  its  own  direc- 
tion is  in  like  manner  caculable  from  a  knowledge  of  that  force,  and 
dividing  this  by  the  linear  velocity  of  P  at  that  instant,  wo  deduce  the 
angular  velocity  of  the  tangent  about  P,  or  the  momentary  variation  of 
the  angle  of  tangency  SPY,  corresponding. 

(G73.)  The  following  rhum6  of  these  several  results  in  a  tabular  form 
includes  every  variety  of  case  according  as  P  is  approaching  to  or  receding 
from  S ;  as  it  is  situated  in  the  arc  FAG  of  its  orbit  about  the  perihe- 
lion, or  in  the  remoter  arc  G  M  F  about  the  aphelion,  as  the  tangential 
force  accelerates  or  retards  the  disturbed  body,  or  as  the  normal  acts  in- 
wards  or  outwards  with  reference  to  the  concavity  of  the  orbit. 


EFFECTS   OF   THE   TANGENTIAL  DISTURBTXG   FORCE. 


^1 


Direction  of  P's  mo- 
tion. 

Situation  of  P  in 
orbit 

Action  of  Tangential 
Force. 

Efiect  on  Elemcnte. 

Approaching  S. 

Ditto. 
Receding  from  S. 

Ditto. 
IndifiFerent. 

Ditto. 

Ditto. 

Ditto. 

Anywhere. 

Ditto. 

Diito. 

Ditto. 
About  Aphelion. 

Ditto. 
About  Perihelion. 

Ditto. 

Accelerating  P. 
Retarding  P. 
Accelerating  P. 
Retarding  P. 
Accelerating  P. 
Retarding  P, 
Accelerating  P. 
Retarding  P. 

Apsides    recede. 

advance. 

advance. 

recede. 
Excentr.  decreases,  j 

increases. 

increases. 

decreases. 

12  11 

w",  and  -r  = w"  ;.  — =  «»— t)'*=(t»+i>')  («—«')  or  when  infi- 

fl  T  #1/1 


a 

a'— a 


,   1        2 

'  —  = v,  ana-r  = 

a       r  a       r  a' 

mtesimal  variations  only  are  considered  - — -^  =  2t;  («' — v)  or   a' — a=2a*t)  (c'— «) 

a' 

from  which  it  appears  that  the  variation  of  the  axis  arising  from  a  given  variation  of 
velooity  is  independent  of  r,  or  is  the  same  at  whatever  distance  from  S  the  change 
takes  place,  and  that  cceteris  paribus  it  is  greater  for  a  given  change  of  velocity  (or  tor 
a  given  tangential  forc(  )  in  the  direct  ratio  of  the  velocity  itself. 


EFFECTS   ON  THE  APSIDES  AND   EXCENTRICITIES. 


EFFECTS  OF  THE   NORMAL  DISTURBING  FORCE. 


369 


1  ■■ 

Direction  of  P**  iao- 
tlon 

Situation  of  P  in 
orbit. 

Action  of  Normal 
Force. 

Effflot  on  Elomertts. 

Indiflerent 

Ditto. 

Ditto. 

Ditto. 
Approaching  8. 

Ditto. 
Receding  from  S, 

Ditto. 

About  Aphelion. 

Ditto. 
About  Perihelion. 

Ditto. 
Anywhere. 

Ditto. 

Ditto. 

Ditto. 

Inwards. 

Outwards. 

Inwards. 

Outwards. 

Inwards. 

Outwards. 

Inwards. 

Outwards. 

Apsides    reoodo. 

advance. 
1        advance.     ' 

recede. 
Excentr.  increases. 

dureases. 

dtcro.'icoH. 

increasi  m. 

(674.)  From  the  momentary  changes  in  the  elements  of  the  disturbed 
orbit  corresponding  to  successive  situations  of  P  and  M,  to  conclude  the 
total  amount  of  change  produced  in  any  given  time  is  the  business  of  the 
integral  calculus,  and  lies  far  beyond  the  scope  of  the  present  work. 
\Vithout  its  aid,  however,  and  by  general  consideratioti  of  the  periodical 
recurrence  of  configurations  of  the  same  character,  we  Lave  been  sble  to 
demonstrate  many  of  the  most  interesting  conclusions  to  which  geometers 
have  been  conducted,  examples  of  which  have  already  been  given  in  tiie 
reasoning  by  which  the  permanence  of  the  axes,  the  periodicity  of  the 
inclinations,  and  the  revolutions  of  the  nodes  of  the  planetary  orbits  have 
been  demonstrated.  We  shall  now  proceed  to  apply  similar  considerations 
to  the  motion  of  the  apsides,  and  the  variations  of  the  excentricities.  To 
this  end  we  must  first  trace  the  changes  induced  on  the  disturbing  forces 
themselves  with  the  varying  positions  of  the  bodies,  and  here  as  in  treat- 
ing of  the  inclinations  we  shall  suppose,  unless  the  contrary  is  expressly 
indicated,  both  orbits  to  be  very  nearly  circular,  without  which  limitation 
the  complication  of  the  subject  would  become  too  embarrassing  for  the 
reader  to  follow,  and  defeat  the  end  of  explanation. 

(675.)  On  this  supposition  the  directions  of  S  P  and  S  Y,  the  perpen- 
dicular on  the  tangent  at  P,  may  be  regarded  as  coincident,  and  the 
normal  and  radial  disturbing  forces  become  nearly  identical  in  quantity, 
also  the  tangential  and  transversal,  by  the  near  coincidence  of  the  points 
T  and  L  (fig.  art.  687).  So  far  then  as  the  intemity  of  the  forces  is  con- 
cerned, it  will  make  very  little  difference  in  which  way  the  forces  are  re- 
solved, nor  will  it  at  all  materially  affect  our  conclusions  as  to  the  effects 
of  the  normal  and  tangential  forces,  if  in  estimating  their  quantitative 
values,  we  take  advantage  of  the  simplification  introduced  into  their  nu- 
merical expression  by  the  neglect  of  the  ingle  P  S  Y,  i.  e.  by  the  substi- 
tution for  them  of  the  radial  and  transversal  components.  The  character 
24 


PHi 

KJ^'-lfiilH' 

HHiK^'              r ' '  '' 

HBEj.             '  ;  M '  < 

'^1 


370 


OUTLINES   OF   ASTRONOMY. 


of  these  effects  depends  (art.  670,  671,)  on  the  direction  in  which  the 
forces  act,  which  we  shall  suppose  normal  and  tangential  as  before,  and  it 
is  only  on  the  estimation  of  their  quantitative  effects  that  the  error  in- 
duced by  the  neglect  of  this  angle  can  fall.  In  the  lunar  orbit  this  angle 
never  exceeds  3°  10',  and  its  influence  on  the  quantitative  estimation  of 
the  acting  forces  may  therefore  be  safely  neglected  in  a  first  approxima- 
tion. Now  M  N  being  found  by  the  proportion  M  P*  :  M  S^ : :  M  S  : 
M  N,  N  P  (=  M  N  —  M  P)  is  also  known,  and  therefore  N  L  =  N  P 
sin  NPS  =  NP.  sin  (ASP+SMP)  and  LS  =  PL— PS  =  NP. 
cos  N  P  S  —  P  3  ^-:  i^  P.  cos  (A  S  P+  S  M  P)  —  S  P  become  known, 
which  express  r<i&pcv lively  the  tangential  and  normal  forces  on  the  saioe 
scale  tliat  &  M  reprejents  M's  attraction  on  S.'  Suppose  P  to  revolve  in 
the  direction  E  A  ' "  B.  Then,  by  drawing  the  figure  in  various  situations 
of  P  throughout  the  whole  circle,  the  reader  will  easily  satisfy  himself  — 
Ist.  That  the  tangential  force  accelerates  P,  as  it  moves  from  E  towards 
A,  and  from  D  towards  B,  but  retards  it  as  it  passes  from  A  to  D,  and 
from  B  to  E.  2d.  That  the  tangential  force  vanishes  at  the  four  points 
A,  D,  E,  B,  and  attains  a  maximum  at  some  intermediate  points. 
3dly.  That  the  normal  force  is  directed  outwards  at  the  syzygies  A,  B, 
and  inwards  at  the  points  D,  E,  at  which  points  respectively  its  outward 
and  inward  intensities  attain  their  maxitua.     Lastly,  that  this  force  va- 


'MS  =  R;SP  =  r;MP=/;   ASP  =  0;  AMP  =  M;  MN- 


1! 


NP  = 


?1_2!  =  (R  _y )  A  -f.  2: ^  LV  whence  we  have  N  L  =  (R  -/).  sin  (9  +  M). 

(l  +  j  +  jl);  LS=-(R-/).  cos(0  +  M).(l  +  -^  +  ^[)-r.    When  R  and/, 

owing  to  the  great  distance  of  M,  are  nearly  equal,  we  have  R  — /=  P  V,  y  =  1 
nearly,  and  the  angle  M  may  be  neglected ;  so  that  we  have  N  P  =  3  P  V. 


DISTURUINO   FORCES   IN  CIRCULAR   ORBITS. 


871 


nishes  at  points  intennediuto  between  A  D,  T)  B,  B  E,  and  E  A,  which 
points,  when  M  is  considerably  remote,  arc  situated  ncaror  to  the  quadra- 
ture than  the  syzygies. 

(07<'»  )  In  the  lunar  theory,  to  which  we  shall  now  proceed  to  apply 
these  principles,  both  the  geometrical  representation  and  the  algebraic 
expression  of  the  disturbing  forces  admit  of  great  simplification.  Owing 
to  the  great  distance  of  the  sun  M,  at  whose  centre  the  radius  of  the 
mcM^n's  orbit  never  subtends  an  angle  of  more  than  about  8',  N  P  may  be 
rognrdod  as  parallel  to  A  B.  And  D  S  E  beconi-  'raipht  line,  coinci- 
dent with  the  line  of  quadratures,  so  that  \ 
ASP  to  radius  S  P,  and  N  L  =  N  P .  siu 
A  S  P.  Moreover,  in  this  case  (see  the  note  (  ' 
3  P  V=  3  S  P .  cos  A  8  P ;  and  consequently  J^  u 
sin  A  S  P  =  :5  S  P.  sin  2  A  S  P,  and  L  S  =  S  P  (3  .  cos  AS  P^--l) 
=  J  S  P  (1  -f  3  .  cos  2  A  S  P)  which  vanishes  when  cos  A  S  P'  =  ^,  or  at 
C4°  14'  from  thn  syzygy.  Suppose  through  every  point  of  P's  orbit 
tlicro  be  drawn  S  Q  =  3  S  P.  cos  A.  S  P'^,  then  will  Q  trace  out  a  certain 
looped  ov!il,  as  in  the  figure,  cutting  tlie  orbit  in  four  points  6-4°  14'  from 
A  and  B  respectively,  and  P  Q  will  dways  represent  in  quantity  and  di- 
rection the  normal  force  acting  at  P. 

Fig.  90. 


:ir'.  the  cosine  of 

=  N  P .  cos 

a-ticle)  N  P  = 

3SP.C0S  ASP. 


;-'^^ 


U 


(077.)  It  is  important, to  remark  here,  because  upon  this  the  whole 
lunar  theory  and  especially  that  of  the  motion  of  the  apsides  hinges,  thai 
all  the  acting  disturbing  forces,  at  equal  angles  of  elongation  A  S  P  of  the 
moon  from  the  sun,  are  ceclcris  jparihus  proportional  to  SP,  the  moon's 
distance  from  the  earth,  and  are  therefore  greater  when  the  moon  is  near 
its  apogee  than  when  near  its  perigee;  the  extreme  proportion  being  that 
of  about  28:25.  This  premised,  let  us  first  consider  the  effect  of  the 
normal  force  in  displacing  the  lunar  apsides.  This  we  shall  best  be  ena- 
bled to  do  by  examining  separately  those  cases  in  which  the  effects  are 


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^m      OUTLINES  OP  ASTRONOMY. 


•^■n 


most  strongly  contrasted ;  viz.  \rhen  the  major  axis  of  the  moon's  orbit  is 
directed  towards  the  sun,  and  when  at  right  angles  to  that  direction. 
First,  then,  let  the  line  of  apsides  be  directed  to  the  sun  as  in  the  an< 
nexed  figure,  where  A  is  the  perigee,  and  take  the  arcs  A  a,  A  &,  B  c,  B  c? 


*^yu??'C.)     ..r:4»i-' 

O'h  ^ti  fHUks^ 

:'' 

lo  '*i'.\mi*  ffih            1 

,  "1^       l  V    .^.  '  +  +  ( 

t^-'l  ^  ''%?*^.-          \ 

S.^'A-9^>^hK^    + 

,   f          i'HC  it    '      .    .V 

&.  "'■<".  it ,  «■  '^'fi  ^-4  ^^^ 

Fig.  91. 


t  '5;-'^ 


■'-'.  . 


eachs64''  14'.  Then  while  P  is  between  a  and  6  the  normal  forc^  act- 
ing outwards,  and  the  moon  being  near  its  perigee,  by  art.  671,  the 
apsides  will  recede,  but  when  between  c  and  d,  the  force  there  acting  out- 
wards, but  the  moon  being  near  its  apogee,  they  will  advance.  The  ra- 
pidity of  these  movements  will  be  respectively  at  its  maxima  at  A  and  B, 
not  only  because  the  disturbing  forces  are  then  most  intense,  but  also 
because  (see  art.  671)  they  act  most  advantageously  at  those  points  to 
displace  the  axis.  Proceeding  from  A  and  B  towards  the  neutral  points 
abed,  the  rapidity  of  their  recess  and  advance  diminishes,  and  is  nothing 
(or  the  apsides  are  stationary)  when  P  is  at  either  of  these  points.  From 
h  to  D,  or  rather  to  a  point  some  little  beyond  D  (art.  671)  the  force  acts 
inwards,  and  the  moon  is  still  near  perigee,  so  that  in  this  arc  of  the  orbit 
the  apsides  advance.  But  the  rate  of  advance  is  feeble,  because  in  the 
early  part  of  that  arc  the  normal  force  is  small,  and  as  P  approaches  D 
and  the  force  gains  power,  it  acts  disadvantageously  to  move  the  axis,  its 
effect  vanishing  altogether  when  it  arrives  beyond  D  at  the  extremity  of 
the  perpendicular  to  the  upper  focus  of  the  lunar  ellipse.  Thence  up  to 
c  this  feeble  advance  is  reversed  and  converted  into  a  recess,  the  force  still 
acting  inwards,  but  the  moon  now  being  near  its  apogee.  And  so  also 
for  the  arcs  dfE,  E  a.     In  the  figure  these  changes  are  indicated  by  +  + 

for  rapid  advance, for  rapid  recess,  +  and  —  for  feeble  advance  and 

recess,  and  0  for  the  stationary  points.  Now  if  tl-e  forces  were  equal  on 
the  sides  of  +  and  —  it  is  evident  that  there  would  be  an  exact  counter- 
balance of  advance  and  recess  on  the  average  of  a  whole  revolution.  But 
this  is  not  the  case.     The  force  in  apogee  is  greater  than  that  in  perigee 


MOTION  OF  THE  LUNAB  APSIDES. 


878 


ia  the  proportion  of  28  :  25,  while  in  the  quadratures  about  D  and  E 
they  are  equal.  Therefore,  while  the  feeble  movements  +  and  —  in  the 
oeigbbourhood  of  these  points  destroy  each  other  almost  exactly,  there 
will  necessarily  remain  a  considerable  balance  in  favour  of  advance,  in 
this  situation  of  the  line  of  apsides. 

(678.)  Next,  suppose  the  apogee  to  lie  at  A,  and  the  perigee  at  B.  In 
this  case  it  is  evident  that,  so  far  as  the  direction  of  the  motions  of  the 
apsides  is  concerned,  all  the  conclusions  of  the  foregoing  reasoning  will 
be  reversed  by  the  substitution  of  the  word  perigee  for  apogee,  and  vice 
versd ;  and  all  the  signs  in  the  figure  referred  to  will  be  changed.  But 
DOW  the  most  powerful  forces  act  on  the  side  of  A,  that  is  to  say,  still  on 
the  side  of  advance,  this  condition  also  being  reversed.  In  either  sitna< 
tioD  of  the  orbit,  then,  the  apsides  advance.  n.%  ^iMi  ■^.';»V 

(679.)  (Case  8.)  Suppose,  now,  the  major  axis  to  have  the  situation 
D  E,  and  the  perigee  to  be  on  the  side  of  D.  Here,  in  the  arc  6  c  of  P's 
motion  the  normal  force  acts  inwards,  and  the  moon  is  near  perigee,  con> 
sequently  the  apsides  advance,  but  with  a  moderate  rapidity,  the  maxi- 
mum of  the  inward  normal  force  being  only  half  that  of  the  outward. 
In  the  arcs  A  b  and  c  B  the  moon  is  still  near  perigee,  and  the  force  acts 
outwards,  but  though  powerfully  towards  A  and  B,  yet  at  a  constantly 
increasing  disadvantage  (art.  671.)  Therefore  in  these  arcs  the  apsides 
recede,  but  moderately.  In  a  A  and  B  d  (being  towards  apogee)  they 
again  advance,  still  with  a  moderate  velocity.  Lastly,  throughout  the  arc 
da,  being  about  apogee  with  an  inward  force,  they  recede.  Here  as 
before,  if  the  perigee  and  apogee  forces  were  equal,  the  advance  and  recess 
would  counterbalance ;  but  as  in  fact  the  apogee  forces  preponderate,  there 
will  be  a  balance  on  the  entire  revolution  in  favour  of  recess.  The  same 
reasoning  of  course  holds  good  if  the  perigee  be  towards  £.  But  now, 
between  these  cases  and  those  in  the  foregoing  articles,  there  is  this  dif- 
ference, viz.  that  in  this  the  dominant  effect  re&alts  from  the  inward  action 
of  the  normal  force  in  quadratures,  while  iu  the  others  it  results  from  its 
outward,  and  doubly  powerful  action  in  syzygies.  The  recess  of  the  ap- 
sides in  their  quadratures  arising  from  the  action  of  the  normal  force  will 
therefore  be  less  than  their  advance  in  their  syzygies ;  and  not  only  on 
this  account,  but  also  because  of  the  much  less  extent  of  the  arcs  h  c  and 
da  on  which  the  balance  is  mainly  struck  in  this  case,  than  oi  ab  and 
cdy  the  corresponding  most  influential  arcs  in  the  other.       .:  i'.'^  (4iht<??Yi 

(680.)  In  intermediate  situations  of  the  line  of  apsides,  the  effect  will 
be  intermediate,  and  there  will  of  course  be  a  situation  of  them  in  which 
on  an  average  of  a  whole  revolution,  they  are  stationary.  This  situation 
it  is  easy  to  see  will  be  nearer  to  the  line  of  quadratures  than  of  syzygies^ 


\\ 


874 


OUTLINES  OP  ASTRONOMY. 


m 


and  the  preponderance  of  advance  will  be  maintained  over  a  much  more 
considerable  arc  than  that  of  recess,  among  the  possible  situations  which 
they  can  hold.  On  every  account,  therefore,  the  action  of  the  normal 
force  causes  the  lunar  apsides  to  progress  in  a  complete  revolution  of  M 
or  in  a  synodical  year,  during  which  the  motion  of  the  sun  round  the 
earth  (as  we  consider  the  earth  at  rest)  brings  the  line  of  syzygies  into  all 
situations  with  respect  to  that  of  apsides. 

(681.)  Let  us  next  consider  the  action  of  the  tangential  force.  And 
as  before  (Case  1.),  supposing  the  perigree  of  the  moon  at  A,  and  the 
direction  of  her  revolution  to  be  A  D  B  E,  the  tangential  force  retards 
her  motion  through  the  quadrant  A  D,  in  which  she  recedes  from  S,  there- 
fore by  art.  670  the  apsides  recede.  Through  D  B  the  force  accelerates, 
while  the  moon  still  recedes,  therefore  they  advance.  Through  B  E  the 
force  retards,  and  the  moon  approaches,  therefore  they  continue  to  advance, 
and  finally  throughout  the  quadrant  E  A  the  force  accelerates,  and  the 
moon  approaches,  therefore  they  recede.  In  virtue  therefore  of  this  force, 
the  apsides  recede,  during  the  description  of  the  arc  E  A  D,  and  advince 
during  D  B  E,  but  the  force  being  in  this  case  as  in  that  of  the  normal 
force  more  powerful  at  apogee,  the  latter  will  preponderate;  and  the  apsides 
will  advance  on  an  average  of  a  whole  revolution.  i«.^»K'  -." ; 

(682.)  (Oase  2.)  The  perigee  being  towards  B,  we  have  to  substitute 
in  the  foregoing  reasoning  approach  to  S,  for  recess  from  it,  and  vice  versd, 
the  accelerations  and  retardations  remaining  as  before.  Therefore  the  re- 
sults, as  far  as  direction  is  concerned,  will  be  reversed  in  each  quadrant, 
the  apsides  advance  during  E  A  D  and  recede  during  D  B  E.  But  the 
situation  of  the  apogee  being  also  revt;r8ed,  the  predominance  remains  on 
the  side  of  E  A  D,  that  is,  of  advanr  -m^-n^-    >.    < 

'  (683.)  (Case  3.)  Apsides  in  quadr  .  js,  perigee  near  D. — Over  qua- 
drant A  D,  approach  and  retardation,  therefore  advance  of  apsides.  Over 
D  B  recess  and  acceleration?  therefore  again  advance;  over  B  E  recess 
and  retardation  with  recess  of  apsides,  and  lastly  over  E  A  approach  and 
acceleration,  producing  their  continued  recess.  Total  result:  advance 
during  the  half  revolution  A  D  B,  and  recess  during  B  E  A,  the  acting 
forces  being  more  pc  v/erful  in  the  latter,  whence  of  course  a  preponderant 
recess.     The  same  result  when  the  perigee  is  at  E. 

(684.)  So  far  the  analogy  of  reasoning  between  the  action  of  the  tan- 
gential and  normal  forces  is  perfect.  But  from  this  point  they  diverge. 
It  is  not  here  as  before.  The  recess  of  the  apsides  in  quadratures  docs 
not  now  arise  from  the  predominance  of  feeble  over  feebler  forces,  while 
that  in  syzygies  results  from  that  of  powerful  over  powerful  ones.  The 
maximum  accelerating  action  of  the  tangential  force  is  equal  to  its  maxi- 


MOTION  OF  THE   LUNAR  APSIDES. 


375 


mum  retarding,  while  the  inward  action  of  the  normal  at  its  maximum  is 
only  half  the  maximum  of  its  outward.  Neither  is  there  that  difference 
in  the  extent  of  the  arcs  over  which  the  balance  is  struck  in  this,  as  in 
the  other  case,  the  action  of  the  tangential  fbroe  being  inward  and  outward 
alternately  over  equal  arcs,  each  a  complete  quadrant.  Whereas,  there- 
fore, in  tracing  the  action  of  the  normal  force,  we  found  reason  to  con- 
clude it  much  more  effective  to  produce  progress  of  the  apsides  in  their 
gyzygy,  than  in  their  quadrature  situations,  we  can  draw  no  such  conclu- 
sion in  that  of  the  tangential  forces :  there  being,  as  regards  that  force,  a 
compkte  tymmetry^  in  the  four  quadrants,  while  in  regard  of  the  normal 
force  the  symmetry  is  only  a  half-tymmetry  having  relation  to  two  temi- 
circles. 

(685.)  Taking  the  average  of  many  revolutions  of  the  sun  about  the 
earth,  in  which  it  shall  present  itself  in  every  possible  variety  of  situations 
to  the  line  of  apsides,  we  see  that  the  effect  of  the  normal  force  is  to  pro- 
duce a  rapid  advance  in  the  syzygy  of  the  apsides,  and  a  less  rapid  recess 
in  their  quadrature,  and  on  the  whole,  therefore,  a  moderately  rapid  gene- 
ral advance,  while  that  of  the  tangential  is  to  produce  an  equally  rapid 
advance  in  syzygy,  and  recess  in  quadrature.  Directly,  therefore,  the 
tangential  force  would  appear  to  have  no  ultimate  influence  in  causing 
either  increase  or  diminution  in  the  mean  Tnotion  of  the  apsides  resulting 
from  the  action  of  the  normal  force.  It  does  so,  however,  indirectly, 
conspiring  in  that  respect  with,  and  greatly  increasing,  an  indirect 
action  of  the  normal  force  in  a  manner  which  we  shall  now  proceed  to 
explain. 

(686.)  The  sun  moving  uniformly,  or  nearly  so,  in  the  same  direction 
as  P,  the  line  of  apsides  when  in  or  near  the  syzygy,  in  advancing  follows 
the  sun,  and  therefore  remiiins  materially  longer  in  the  neighbourhood  of 
syzygy  than  if  it  rested.  On  the  other  hand,  when  the  apsides  are  in 
quadrature  they  recede,  and,  moving  therefore  contrary  to  the  sun's 
motion,  remain  a  shorter  time  in  that  neighbourhood,  thaa  if  they  rested. 
Thus  the  advance,  already  preponderant,  is  made  to  preponderate  more 
by  its  longer  continuance,  and  the  recess,  already  deficient,  is  rendered 
still  more  so  by  the  shortening  of  its  duration.'  Whatever  cause,  then, 
increases  directly  the  rapidity  of  both  advance  and  recess,  though  it  may 
do  both  expially,  aids  in  this  indirect  process,  and  it  is  thus  that  the  tan- 
gential force  becomes  effective  through  the  medium  of  the  progress  already 
produced,  in  doing  and  aiding  the  normal  force  to  do  that  which  alone  it 
would  be  unable  to  effect.  Thus  we  have  perturbation  exaggerating 
perturbation,  and  thus  we  see  what  is  meant  by  geometers,  when  they 

•Newton,  Princ.  i.  66.  Cor.  8. 


876 


/tK 


OUTLINES  OF  ASIBONOMT. 


li 


declare  that  a  considerable  part  of  the  motion  of  the  lunar  apsides  is  due 
to  the  square  of  the  disturbing  force,  or,  in  other  words,  arises  out  of  a 
second  approximation  in  which  the  influence  of  the  first  in  altering  the 
data  of  the  problem  is  taken  into  account. 

(687.)  The- curious  and  complicated  effect  of  perturbation,  described  in 
the  last  article,  has  given  more  trouble  to  geometers  than  any  other  part 
of  the  lunar  theory.  Newton  himself  had  succeeded  in  tracing  that  part 
of  the  motion  qi  the  apogee  which  is  due  to  the  direct  action  of  the  radial 
force;  but  finding  the  amount  only  half  what  observation  assigns,  he 
appears  to  have  abandoned  the  subject  in  despair.  Nor,  when  resumed 
by  his  successors,  did  the  inquiry,  for  a  very  long  period,  assume  a  more 
promising  aspect.  On  the  contrary,  Newton's  result  appeared  to  be  even 
minutely  verified,  and  the  elaborate  investigations  which  were  lavished 
upon  the  subject  without  success,  began  to  excite  strong  doubts  whether 
this  feature  of  the  lunar  motions  could  be  explained  at  all  by  the  New- 
tonian law  of  gravitation.  The  doubt  was  removed,  however,  almost  in 
the  instant  of  its  origin,  by  the  same  geometer,  Glairaut,  who  first  gatve  It 
currency,  and  who  gloriously  repaired  the  error  of  his  momentary  hesita- 
tion, by  demonstrating  the  exact  coincidence  between  theory  and  observa- 
tion, when  the  effect  of  the  tangential  force  is  properly  taken  into  the 
account.  The  lunar  apogee  circulates,  in  3232''-575343,  or  about  9} 
years. 

(688.)  Let  us  now  proceed  to  investigate  the  influence  of  the  disturbing 
forces  so  resolved  on  the  excentricity  of  the  lunar  orbit,  and  the  foregoing 
articles  having  sufficiently  familiarized  the  reader  with  our  mode  of  fol- 


Fig.  92. 


"•( 


'ft.  •*.--;; 


M 


\^ 


M>;  ;,. 


lowing  out  the  changes  in  different  situations  of  the  orbit,  we  shall  take 
at  once  a  more  general  situation,  and  suppose  the  line  of  apsides  in  any 
position  with  respect  to  the  sun,  such  as  Z  Y,  the  perigee  being  at  Z,  a 
point  between  the  lower  syzygy  and  the  quadrature  next  foUowiDg  it,  the 
direction  of  P's  motion  as  all  alone  supposed  being  A  D  B  E.     Then 


VARIATION   OF  IHB  MOON'S  EXOBNIRTOITT. 


877 


(oommeDciog  with  the  normal  force)  the  momentary  change  of  ezcentri- 
city  will  vanish  at  a,  b,  c,  d,  by  the  vanishing  of  that  force,  and  at  Z  and 
Y  by  the  effect  of  situation  in  the  orbit  annulling  its  action  (art.  671). 
In  the  arcs  Z  b  and  Y  d  therefore  the  change  of  excentricity  will  be  small, 
the  acting  force  nowhere  attaining  either  a  great  magnitude  or  an  advan- 
tageous situation  within  their  limits.  And  the  force  within  these  two 
arcs  having  the  same  character  as  to  inward  and  outward,  but  being  oppo- 
sitely influential  by  reason  of  the  approach  of  F  to  S  in  one  of  them  and 
its  recess  in  the  other,  it  is  evident  that,  so  far  as  these  arcs  are  concerned, 
a  very  near  compensation  of  effects  will  take  place,  and  though  the  apo- 
geal  arc  Y  d  will  be  somewhat  more  influential,  this  will  tell  for  little 
upon  the  average  of  a  revolution. 

(689.)  The  arcs  bJ)  c  and  <2  E  a  are  eaeh  much  less  than  a  quadrant 
in  extent,  and  the  force  acting  inwards  throughout  them  (which  at  its 
maxiTuum  in  D  and  E  is  only  half  the  outward  force  at  A,  B)  degrades 
very  rapidly  in  intensity  towards  either  syzygy  (see  art.  676).  Hence 
whether  Z  be  between  bcorbAf  the  effects  of  the  force  in  these  arcs 
will  not  produce  very  extensive  changes  on  the  excentricity,  and  the 
changes  wh^ch  it  does  produce  will  (for  the  reason  already  given)  be  op- 
posed to  each  other.  Although,  then,  the  arc  ad  he  farther  from  perigee 
than  b  c,  and  therefore  the  force  in  it  is  greater,  yet  the  predominance  of 
effect  here  will  not  be  very  marked,  and  will  moreover  be  partially  neu- 
tralized by  the  small  predominance  of  an  opposite  character  in  Y  (2  over 
Z  b.  On  the  other  hand,  the  arcs  a  Z,  c  Y  arc  both  larger  in  extent  than 
either  of  the  others,  and  the  seats  of  action  of  forces  doubly  powerful. 
Their  influence,  therefore,  will  be  of  most  importance,  and  their  prepon- 
derance one  over  the  other  (being  opposite  in  their  tendencies),  will  decide 
the  question  whether,  on  an  average  of  the  revolution,  the  excentricity 
shall  increase  or  diminish.  It  is  clear  that  the  decision  must  be  in  favour 
of  c  Y,  the  apogeal  arc,  and,  since  in  this  the  force  is  outwards  and  the 
moon  receding  from  the  earth,  an  increase  of  the  excentricity  will  arise 
from  its  influence.  A  similar  reasoning  vdll,  evidently,  lead  to  the  same 
conclusion  were  the  apogee  and  perigee  to  change  places,  for  the  directions 
of  P's  motion  as  to  approach  and  recess  to  S  will  be  indeed  reversed,  but 
at  the  same  time  the  dominant  forces  will  have  changed  sides,  and  the 
arc  a  A  Z  will  now  give  the  character  to  the  result.  But  when  Z  lies 
between  A  and  E,  as  the  reader  may  easily  satisfy  himself,  the  case  will 
be  altogether  different,  and  the  reverse  conclusion  will  obtain.  Hence  the 
changes  of  excentricity  emergent  on  the  average  of  single  revolutions 
from  the  action  of  the  normal  force  will  be  as  represented  by  the  signs 
+  and  —  in  the  figure  above  annexed. 


I 
I 


878 


;''»    OUTLINES   OP  ASTRONOMY.     '*i«i' 


(690.)  Let  UB  next  consider  the  effect  of  the  tangential  force.    This 
retards  P  in  the  qaadrants  A  D,  B  E,  and  accelerates  it  in  the  alternate 


.lU'-^-  ■■ 

■ 

Fig. 

98. 

^r/n.  ':  . 

•J  ' 

D 

■i'l "  • 

.ru'l  ;;'■;'/;;■  ,::.< 

)./,       v.«„^\.           X^ 

^ 

,•■    :,'(Vi;?^    i;                    ■r'^. 

Z 

/ 

\ 

'■'*',*.     3  ,•■* »       \           > 

7 

J 

/       .,.„  ,^    ^^i'^t''  -^^A;.^ 

iy^y:    -vi-''    >;:• 

:« 

^ 

y 

1  i.;<^i-       >,;;;.•»<»*■.:     ;■ 

ones.  In  the  whole  quadrant  A  D,  therefore,  the  effect  is  of  one  charao 
ter,  the  perigee  being  less  than  90°  from  every  point  in  it,  and  in  the 
whole  quadrant  B  E  it  is  of  the  opposite,  the  apogee  being  so  situated 
(art.  670).  Moreover,  in  the  middle  of  each  quadrant,  the  tangential 
force  is  at  its  maximum.  Now,  in  the  other  quadrants,  E  A  and  V  B, 
the  change  from  perigeal  to  apogeal  vicinity  takes  place,  and  the  tangen- 
tial force,  however  powerful,  has  its  effect  annulled  by  situation  (art.  670), 
and  this  happens  more  or  less  nearly  about  the  points  where  the  force  is  a 
maximum.  These  quadrants,  then,  are  far  less  influential  on  the  total 
result,  so  that  the  character  of  that  result  will  be  decided  by  the  predo- 
minance of  one  or  other  of  the  former  quadrants,  and  will  lean  to  that 
which  has  the  apogee  in  it.  Now  in  the  quadrant  B  E  the  force  retards 
the  moon,  and  the  moon  is  in  apogee.  Therefore  the  excentricity  in- 
creases. In  this  situation  therefore  of  the  apogee,  stick  is  the  average 
result  of  a  complete  revolution  of  the  moon.  Here  again  also,  if  the 
perigee  and  apogee  change  places,  so  will  also  the  character  of  all  the  par- 
tial influences,  arc  for  arc.  But  the  quadrant  A  D  will  now  preponderate 
instead  of  D  E,  so  that  under  this  double  reversal  of  conditions  the  result 
will  bo  identical.  Lastly,  if  the  line  of  apsides  be  in  A  E,  B  D,  it  may 
be  shown  in  like  manner  that  the  excentricity  will  diminish  on  the  average 
of  a  revolution. 

(691.)  Thus  it  appears,  that  in  varying  the  excentricity,  precisely  as  in 
moving  the  line  of  apsides,  the  direct  effect  of  the  tangential  force  con- 
spires with  that  of  the  normal,  and  tends  to  increase  the  extent  of  the 
deviations  to  and  fro  on  either  side  of  a  mean  value  which  the  varying 
situation  of  the  sun  with  respect  to  the  line  of  apsides  gives  rise  to,  having 
for  their  period  of  restoration  a  synodical  revolution  of  the  sun  and  apse. 
Supposing  the  sun  and  apsis  to  start  together,  the  sun  of  course  will 
outrun  the  apsis  (whose  period  is  nine  years,)  and  in  the  lapse  of  about 


EXPERIMENTAL  ILLUSTRATION. 


879 


(}+i\z)  P*^  °^  '^  year  will  have  gained  on  it  90°,  during  all  which  inter- 
val the  apse  will  have  heen  in  the  quadrant  A  E  of  our  figure,  and  the 
ezcentricity  continually  decreasing.  The  decrease  will  then  cease,  but 
the  excentrioity  itself  will  be  a  minimum,  the  sun  being  now  at  right 
angles  to  the  line  of  apsides.  Thence  it  will  increase  to  a  maximum 
when  the  sun  has  gained  another  90",  and  again  attained  the  line  of 
apsides,  and  so  on  alternately.  The  actual  effect  on  the  numerical  value 
of  the  lunar  excentricity  is  very  considerable,  the  greatest  and  least  excen- 
tricities  being  in  the  ratio  of  3  to  2.' 

(692.)  The  motion  of  the  apsides  of  the  lunar  orbit  may  be  illustrated 
by  a  very  pretty  mechanical  experiment,  which  is  otherwise  instructive  in 
giving  an  idea  of  the  mode  in  which  orbitual  motion  is  carried  on  under 
the  action  of  central  forces  variable  according  to  the  situation  of  the 
revolving  body.  Let  a  leaden  weight  be  suspended  by  a  brass  or  iron 
wire  to  a  hook  in  the  under  side  of  a  firm  beam,  so  as  to  allow  of  its  free 
motion  on  all  sides  of  the  vertical,  and  so  that  when  in  a  state  of  rest  it 
shall  just  clear  the  floor  of  the  room,  or  a  table  placed  ten  or  twelve  feet 
beneath  the  hook.  The  point  of  support  should  be  well  secured  from 
wagging  to  and  fro  by  the  oscillation  of  the  weight,  which  should  be 
sufficient  to  keep  the  wire  as  tightly  stretched  as  it  will  bear,  with  the 
certainty  of  not  breaking.  Now,  let  a  very  small  motion  be  communi* 
cated  to  the  weight,  not  by  merely  withdrawing  it  from  the  vertical  and 
letting  it  fall,  but  by  giving  it  a  slight  impulse  sideways.  It  will  be  seen 
to  describe  a  regular  ellipse  about  the  point  of  rest  as  its  centre.  If  the 
wcigut  be  heavy,  and  carry  attached  to  it  a  pencil,  whose  point  lies 
exactly  in  the  direction  of  the  string,  the  ellipse  may  be  transferred  to 
paper  lightly  stretched  and  gently  pressed  against  it.  In  these  circum- 
stances, the  situation  of  the  major  and  minor  l  r. :  of  the  ellipae  will 
remain  for  a  long  time  very  nearly  the  same,  though  che  resistance  of  the 
air  and  the  stiffness  of  the  wire  will  gradually  diminish  its  dimensions  and 
excentricity.  But  if  the  impulse  communicated  to  the  weight  be  con- 
siderable, so  as  to  carry  it  out  to  a  great  angle  (15"  or  20"  from  the 
vertical,)  this  permanence  of  situation  of  the  ellipse  will  no  longer  subsist. 
Its  axis  will  be  seen  to  shift  its  position  at  every  revolution  of  the  weight, 
advancing  in  the  same  direction  with  the  weight's  motion,  by  an  uniform 
and  regular  progression,  which  at  length  will  entirely  reverse  its  situation, 
bringing  the  direction  of  the  longest  excursions  to  coincide  with  that  in 
which  the  shortest  were  previously  made ;  and  so  on,  round  the  whole 
circle ;  and,  in  a  word,  imitating  to  the  eye,  very  completely,  the  motion 
of  the  apsides  of  the  moon's  orbit. 

*  Airy,  Gravitation,  p.  106. 


^< 


880 


OUTLINES  OF  ASTRONOMT. 


(698.)  Nov,  if  we  inquiro  into  the  oauae  of  this  progroHsioa  of  tbo 
apsides,  it  will  not  be  difficult  of  detection.  When  a  weight  is  suspended 
by  a  wire,  and  drawn  aside  from  the  verticpl,  it  is  urged  to  the  lowest 
point  (or  rather  in  a  direction  at  every  instant  perpendicular  to  the  w^re) 
by  a  force  which  varies  as  the  sine  of  the  deviation  of  the  wire  from  the 
perpendicular.  Now,  the  sines  of  very  small  arcs  are  nearly  in  the  pro« 
portion  of  the  arcs  themselves;  and  the  more  nearly,  as  the  arcs  are 
smaller.  If,  therefore,  the  deviations  from  the  vertical  be  so  small  that 
wo  may  neglect  the  curvature  of  the  spherical  surfifuie  in  which  the  weight 
moves,  and  regard  the  curve  described  as  coincident  with  its  projection  on 
a  horizontal  plane,  it  will  be  then  moving  under  the  same  circumstances 
as  if  it  were  a  revolving  body  attracted  to  a  centre  by  a  force  varying 
directly  as  the  distance ;  and,  in  this  case,  the  curve  described  would  be 
an  ellipse,  having  its  centre  of  attraction  not  in  the  focus,  but  in  the 
centre',  and  the  apsides  of  this  ellipse  would  remain  fixed.  But  if  the 
excursions  of  the  weight  from  the  vertical  be  considerable,  the  force  urging 
it  towards  the  centre  will  deviate  in  its  law  from  the  simple  ratio  of  the 
distances ;  being  as  the  8ine,  while  the  distances  are  as  the  arc.  Now  the 
sine,  though  it  continues  to  increase  as  the  arc  increases,  yet  docs  not  in- 
crease so  fast.  So  soon  as  the  arc  has  any  sensible  extent,  the  sine  begins 
to  full  somewhat  short  of  the  magnitude  which  an  exact  numerical  propor- 
tiooality  would  require;  and  thc^ir^fore  the  foi;ce  urging  the  weight  towards 

■I  Fig.  94. 


Vl    ." 


*  •  r  ■ 


its  centre  or  point  of  rest  at  great  distances  falls,  in  like  proportion,  some- 
what short  of  that  which  would  keep  the  body  in  its  precise  elliptic  orbit. 
It  will  no  longer,  therefore,  have,  at  those  greater  distances,  the  same 
command  over  the  weight,  in  proportion  to  its  speed,  which  would  enable 
it  to  deflect  it  from  its  rectilinear  tangential  course  into  an  ellipse.  The 
true  path  which  it  describes  will  be  less  curved  in  the  remoter  parts  than 
is  consistent  with  the  elliptic  figure,  as  in  the  annexed  cut;  and,  therefore, 
it  will  not  so  soon  have  its  motion  brought  to  be  again  at  right  angles  to 


*  Newton,  Princi^.  i.  47. 


ilys 


APPLICATION  OF  THB  PLANETARY  THEORY. 


881 


the  radiufl.  It  will  require  a  longer  continued  action  of  the  central  force 
to  do  this ;  and  before  it  is  accomplished,  more  than  a  quadrant  of  its 
revolution  must  he  passed  over  in  angular  motion  round  the  centre.  But 
this  is  only  stating  at  length,  and  in  a  more  circuitous  manner,  that  fact 
which  is  more  briefly  and  summarily  expressed  by  saying  that  the  apnde$ 
of  its  orbit  are  progrenive.  Nothing  beyond  a  familiar  illustration  is  of 
course  intended  in  what  is  above  said.  The  case  is  not  an  exact  parallel 
with  that  of  the  lunar  orbit,  the  disturbing  force  being  simply  radial, 
whereas  in  the  lunar  orbit  a  transversal  force  is  also  concerned,  and  even 
were  it  otherwise,  only  a  confused  and  indistinct  view  of  aspidal  motion 
can  be  obtained  from  this  kind  of  consideration  of  the  curvature  of  the 
disturbed  path.  If  we  would  obtain  a  clear  one,  the  two  foci  of  the  in< 
gtantaneous  ellipse  must  be  found  from  the  laws  of  elliptic  motion  per- 
formed under  the  influence  of  a  force  directly  as  the  distance,  and  the 
radial  disturbing  force  being  decomposed  into  its  tangential  and  normal 
components,  the  momentary  influence  of  either  in  altering  their  positions 
and  consequently  the  directions  and  lengths  of  the  axis  of  the  ellipse 
must  be  ascertained.  The  student  will  find  it  neither  a  difficult  nor  an 
uninstructive  exercise  to  work  out  the  ease  from  these  principles,  which 
we  cannot  afford  the  space  to  do.         '  i      i 

(694.)  The  theory  of  the  motion  of  the  planetary  apsides  and  the 
variation  of  their  excentricities  is  in  one  point  of  view  much  more  simple, 
but  in  another  much  more  complicated  than  that  of  the  lunar.  It  is 
Eimpler,  because  owing  to  the  exceeding  minuteness  of  the  changes  ope- 
rated in  the  course  of  a  single  revolution,  the  angular  position  of  the 
bodies  with  respect  to  the  line  of  apsides  is  very  little  altered  by  the 
motion  of  the  apsides  themselves.  The  line  of  apsides  neither  follows 
up  the  motion  of  the  disturbing  body  in  its  state  of  advance,  nor  vice 
versd,  in  any  degree  capable  of  prolonging  materially  their  advancing  or 
shortening  materially  their  receding  phase.  Hence  no  second  approxima- 
tion of  the  kind  explained  (in  art.  686),  by  which  the  motion  of  the  lunar 
apsides  is  so  powerfully  modified  as  to  be  actually  doubled  in  amount,  is 
at  all  required  in  the  planetary  theory.  On  the  other  hand,  the  latter 
theory  is  rendered  more  complicated  than  the  former,  at  least  in  the  cases 
of  planets  whose  periodic  times  are  to  each  other  in  a  ratio  much  less  than 
13  to  1,  by  the  consideration  that  the  disturbing  body  shifts  its  position 
with  respect  to  the  line  of  apsides  by  a  much  greater  angular  quantity  in 
a  revolution  of  the  disturbed  body  than  in  the  case  of  the  moon.  In  that 
case  we  were  at  liberty  to  suppose  (for  the  sake  of  explanation),  without 
any  very  egregious  error,  that  the  sun  held  nearly  a  fixed  position  during 
a  single  lunation.    But  in  the  case  of  planets  vhose  times  of  revolution 


382 


f' 


OUTLINES  OF  ASTRONOMY.     '  I  M  i  S 


arc  in  a  much  lower  ratio  tbia  oanDOt  bo  pernuttod.  Id  the  case  of  Jupiter 
disturbed  by  Saturn  for  example,  in  one  sidereal  revolution  of  Jupiter, 
Saturn  has  advanced  in  its  orbit  with  respect  to  the  line  of  apsides  of 
Jupiter  by  more  than  140°,  a  change  of  direction  which  entirely  alters 
the  conditions  under  which  the  disturbing  forces  act.  And  in  the  case 
of  an  exterior  disturbed  by  an  interior  planet,  the  situation  of  the  latter 
with  respect  to  the  lino  of  the  apsides  varies  even  more  rapidly  than  tbe 
situation  of  the  exterior  or  disturbed  plitnot  with  respect  to  the  central 
body.  To  such  oases  then  the  reasoning  which  we  have  applied  to  the 
lunar  perturbatisns  becomes  totally  inapplicable ;  and  when  we  take  into 
oonaideration  also  the  excentrioity  of  the  orbit  of  the  disturbing  body, 
which  in  the  most  important  cases  is  exceedingly  influential,  the  subject 
becomes  far  too  complicated  for  verbal  explanation,  and  can  only  be  suo- 
oessfully  followed  out  with  the  help  of  algebraic  expression  and  the  appli« 
cation  of  the  integral  calculus.  To  Mercury,  Venus,  and  the  earth  indeed, 
as  disturbed  by  Jupiter,  and  planets  superior  to  Jupiter,  this  objection  to 
the  reasoning  in  question  does  not  apply;  and  in  each  of  these  4a&cs 
therefore  wo  are  entitled  to  conclude  that  tbe  apsides  are  kept  in  a  state 
of  progression  by  the  action  of  all  the  larger  planets  of  our  system. 
Under  certain  conditions  of  distance,  excentrioity,  and  relative  situation 
of  the  axes  of  the  orbits  of  the  disturbed  and  disturbing  planets,  it  is 
perfectly  possible  that  the  reverse  may  happen,  an  instance  of  which  is 
afforded  by  Venus,  whose  apsides  recede  under  the  combined  action  of  the 
earth  and  Mercury  moro  rapidly  than  they  advance  under  the  joint  actions 
of  all  the  other  planets.  Nay,  it  is  even  possible  under  certain  conditions 
that  the  line  of  apsides  of  the  disturbed  pUnct,  instead  of  revolving  always 
in  one  direction,  may  librate  to  and  fro  within  assignable  limits,  and  in  a 
definite  and  regularly  recurring  period  of  time. 

.  (695.)  Under  any  conditions,  however,  as  to  these  particulars,  the 
view  we  have  above  taken  of  the  subject  enables  us  to  assign  at  every 
instant,  and  in  every  configuration  of  the  two  planets,  the  momentary 
effect  of  each  upon  the  perihelion  and  excentrioity  of  the  other.  In  the 
simplest  case,  that  in  which  the  two  orbits  are  so  nearly  circular,  that 
the  relative  situation  of  their  perihelia  shall  produce  no  appreciable  differ- 
ence in  the  intensities  of  the  disturbing  forces,  it  is  very  easy  to  show 
that  whatever  temporary  oscillations  to  and  fro  in  the  positions  of  the  line 
of  apsides,  and  whatever  temporary  increase  or  diminution  in  the  excen- 
trioity of  either  planet  may  take  place,  the  final  effect  on  the  average  of 
a  great  multitude  of  revolutions,  presenting  them  to  each  other  in  all 
possible  configurations,  must  be  nt7,  for  both  elements. 

(696.)  To  show  this,  all  that  is  necessary  is  to  cast  our  eyes  on  the 


EFFECTS  OF  ELLIPTICITY. 


888 


lynoptio  tablo  in  art.  078.  If  M,  the  disturbing  body,  bo  supposed  to  be 
8UC0088ively  placed  in  two  diainotrically  opposite  situations  io  its  orbit,  the 
aphelioo  of  P  will  stand  related  to  M  in  one  of  these  situations  precisely 
as  its  periholioD  in  the  other.  Now  the  crAla  being  so  nearly  circles  m 
supposed,  the  distribution  of  the  disturbing  forces,  whether  normal  or 
tangential,  is  symmetrical  relative  to  their  common  diameter  passing 
through  M,  or  to  the  line  of  syzygios.  Ilonce  it  follows  that  the  half  of 
P'a  orbit  "  about  perihelion"  (art.  673)  will  stand  related  to  all  the  acting 
forces  in  the  one  situation  of  M,  precisely  as  the  half  "about  aphelion" 
docs  in  the  other :  and  also,  that  the  half  of  the  orbit  in  which  P  *<  ap* 
proacheg  S,"  stands  related  to  them  in  the  one  situation  precisely  as  the 
Lalf  in  which  it  "recedes  from  S"  in  the  other.  Whether  as  regards, 
therefore,  the  normal  or  tangential  force,  the  conditions  of  advance  or 
recess  of  apsides,  and  of  increase  or  diminution  of  ezcentricities,  are 
reversed  in  the  two  supposed  cases.  Hence  it  appears  that  whatever 
situation  be  assigned  to  M,  and  whatever  influence  it  may  exert  on  P  in 
that  situation,  that  influence  will  be  annihilated  in  situations  of  M  and 
F,  diametrically  opposite  to  those  supposed,  and  thus,  ou  a  general 
average,  the  effect  on  both  apsides  and  ezcentricities  is  reduced  to 
nothing. 

(697.)  If  the  orbits,  however,  be  ezcentrio,  the  symmetry  above  in- 
sisted on  in  the  distribution  of  the  forces  does  not  exist.  But,  in  the  first 
place,  it  is  evident  that  if  the  excentricities  be  moderate,  (as  in  the  planet- 
ary orbits,)  by  far  the  larger  part  of  the  effects  of  the  disturbing  forces 
destroys  itself  in  the  manner  described  in  the  lost  article,  and  that  it  is 
only  a  residual  portion,  viz.  that  which  arises  from  the  greater  proximity 
of  the  orbits  at  one  place  than  at  another,  which  can  tend  to  produce  per- 
manent or  secular  effects.  The  precise  estimation  of  these  effects  is  too 
complicated  an  affair  for  us  to  enter  upon ;  but  we  may  at  least  give  some 
idea  of  the  process  by  which  they  are  produced,  and  the  order  in  which 
they  arise.  In  so  doing,  it  is  necessary  to  distinguish  between  the  effects 
of  the  normal  and  tangential  forces.  The  effects  of  the  former  arc  greatest 
at  the  point  of  conjunction  of  the  planets,  because  the  normal  force  itself 
is  there  always  at  its  maximum ;  and  although,  where  the  conjunction 
takes  place  at  90°  from  the  line  of  apsides,  its  effect  to  move  the  apsides 
is  nullified  by  situation,  and  when  in  that  line  its  effect  on  the  excentri- 
cities is  similarly  nullified,  yet,  in  the  situations  rectangular  to  these,  it 
acts  to  its  greatest  advantage.  On  the  other  hand,  the  tangential  force 
vanishes  at  conjunction,  whatever  be  the  place  of  conjunction  with  respect 
to  the  line  of  apsides,  and  where  it  is  at  its  maximum  its  effect  Is  still 
liable  to  be  annulled   by  situation.     Thus  it  appears  that  the  normal 


884 


OUTLINES  OF  ASTRONOMY. 


force  is  most  iofluential,  and  mainly  determines  the  character  of  the  ge- 
neral effect.  It  is,  therefore,  at  conjunction  that  the  most  influential 
effect  is  produced,  and  therefore,  on  the  long  average,  those  conjunctions 
which  happen  about  the  place  where  the  orbits  are  nearest  will  determine 
the  general  character  of  the  effect.  Now,  the  nearest  points  of  approach 
of  two  ellipses,  which  have  a  common  focus,  may  be  variously  situated 
with  respect  to  the  perihelion  of  either.  It  may  be  at  the  perihelion  or 
the  aphelion  of  the  disturbed  orbit,  or  in  any  intermediate  position.  Sup- 
pose  it  to  be  at  the  perihelion.  Then,  if  the  disturbed  orbit  be  interior 
to  the  disturbing,  the  force  acts  outwards,  atid  therefore  the  apsides  re- 
cede :  if  exterior,  the  force  acts  inwards,  and  they  advance.  In  neither 
case  does  the  excentricity  change.  If  the  conjunction  take  place  at  the 
aphelion  of  the  disturbed  orbit,  the  effects  will  be  reversed :  if  interme- 
diate, the  apsides  will  be  less,  and  the  excentricity  more  affected. 

(698.)  Supposing  only  two  planets,  this  process  would  go  on  till  the 
apsides  and  excentricities  had  so  far  changed  as  to  alter  the  point  of 
nearest  approach  of  the  orbits,  so  as  either  to  accelerate  or  retard  land 
perhaps  reverse  the  motion  of  the  apsides,  and  give  to  the  variation  of  the 
excentricity  a  corresponding  periodical  character.  But  there  are  many 
planets,  all  disturbing  one  another.  And  this  gives  rise  to  variations  in 
the  points  of  nearest  approach  of  all  the  orbits,  taken  two  and  two  toge- 
ther, of  a  very  complex  nature. 

(699.)  It  cannot  fail  to  have  been  remarked,  by  any  one  who  has  fol- 
lowed attentively  the  above  reasonings,  that  a  close  analogy  subsists  between 
two  sets  of  relations ;  viz.  that  between  the  inclinations  and  nodes  on  the 
one  hand,  and  between  the  excentricity  and  apsides  on  the  other.  In  fact, 
the  strict  geometrical  theories  of  the  two  cases  present  a  close  analogy, 
and  load  to  final  results  of  the  very  same  nature.  What  the  variation  of 
excentricity  is  to  the  motion  of  the  perihelion,  the  change  of  inclination 
is  to  the  motion  of  the  node.  In  either  case  the  period  of  the  one  is  also 
the  period  of  the  other;  and  while  the  perihelia  describe  considerable 
angles  by  an  oscillatory  motion  to  and  fro,  or  circulate  in  immense  periods 
of  time  round  the  entire  circle,  the  excentricities  increase  and  decrease  by 
comparatively  small  changes,  and  are  at  length  restored  to  their  original 
magnitudes.  In  th^  lunar  orbit,  as  the  rapid  rotation  of  the  nodes  pre- 
vents the  change  of  inclination  from  accumulating  to  any  material 
amount,  so  the  still  more  rapid  revolution  of  its  apogee  effects  a  speedy 
compensation  in  the  fluctuations  of  its  excentricity,  and  never  suffers  thtm 
to  go  to  any  material  extent ;  while  the  same  causes,  by  presenting  in 
qxiick  succession  the  lunar  orbit  in  every  possible  situation  to  all  the  dis- 
turbing forces,  whether  of  the  sun,  the  planets,  or  the  protuberant  matter 


,'f^--' 


V 


OOMPOUND   CYCLES  OF  EXCENIKICITIES,  ETC. 


385 


at  th«7  earth's  equator,  prevent  any  secular  accumulation  of  small  changes, 
by  which,  in  the  lapse  of  ages,  its  ellipticity  might  be  materially  increased 
or  diminished.  Accordingly,  observation  shows  the  mean  excentricity  of 
the  moon'c  orbit  to  be  the  same  now  as  in  the  earliest  ages  of  astronomy. 

(700.)  The  movements  of  the  perihelia,  and  variations  of  excentricity 
nf  the  planetary  orbits,  are  interlaced  and  complicated  together  in  the 
same  manner  and  nearly  by  the  same  laws  as  the  variations  of  their  nodes 
and  inclinations.  Each  acts  upon  every  other,  and  every  such  mutual 
action  generates  its  own  peculiar  period  of  circulation  or  compensation ; 
and  every  such  period,  in  pursuance  of  the  principles  of  art.  650,  is 
thence  propagated  throughout  the  system.  Thus  arise  cycles  upon  cycles, 
of  whose  compound  duration  some  notion  may  be  formed,  when  we  con- 
sider what  is  the  length  of  one  such  period  in  the  case  of  the  two  prin- 
cipal planets  —  Jupiter  and  Saturn.  Neglecting  the  action  of  the  rest, 
*ho  effect  of  their  mutual  attraction  would  be  to  produce  a  secular  varia- 
tion in  the  excentricity  of  Saturn's  orbit,  from  008409,  its  maximum, 
to  001345,  its  minimum  value :  while  that  of  Jupiter  would  vary  be- 
tween the  narrow  limits,  006036  and  0-02606  :  the  greatest  excentricity 
of  Jupiter  corresponding  to  the  least  of  Saturn,  and  vice  versd.  The 
period  in  which  these  changes  are  gone  through,  would  be  70414  years. 
After  this  example,  it  will  be  easily  conceived  that  many  millions  of  years 
will  require  to  elapse  before  a  complete  fulfilment  of  the  joint  cycle  which 
shall  restore  the  whole  system  to  its  original  state  as  far  as  the  exoentri- 
cities  of  its  orbits  are  concerned. 

(701.)  The  place  of  the  perihelion  of  a  planet's  orbit  is  of  little  con- 
sequence to  its  well-being;  but  its  excentricity  is  most  important,  as  upon 
this  (the  axes  of  the  orbits  being  permanent)  depends  the  mean  tempera- 
ture of  its  surface,  and  the  extreme  variations  to  which  its  seasons  may 
be  liable.  For  it  may  be  easily  shown  that  the  m£an  annual  amount 
of  light  and  heat  received  by  a  planet  from  the  sun  is,  cseteris  paribus, 
as  the  minor  axis  of  the  ellipse  described  by  it.  Any  variation,  there- 
fore, in  the  excentricity,  by  changing  the  minor  axis,  will  alter  the  mean 
temperature  of  the  surface.  How  such  a  change  will  also  influence  the 
extremes  of  temperature  appears  from  art.  368.  Now  it  may  naturally 
be  inquired  whether  (in  the  vast  cycle  above  .spoken  of,  in  which,  at  some 
period  or  other,  conspiring  changes  may  accumulate  on  the  orbit  of  one 
planet  from  several  quarters,)  it  may  not  happen  that  the  excentricity  of 
any  one  planet  —  as  the  earth  —  may  become  exorbitantly  great,  so  as  to 
subvert  those  relations  which  render  it  habitable  to  man,  or  to  give  rise  to 
great  changes,  at  least,  in  the  physical  comfort  of  his  state.  To  this  the 
researches  of  geometers  have  enabled  us  to  answer  in  the  negative.  A 
26 


':',;';.•  ■^'•^^ig"'''^: 


i 


886 


OUTLINES  OF  ASIRONOMT. 


relation  has  been  demonstrated  by  Lagrange  between  the  masses,  axes  of 
the  orbits,  and  ezcentricities  of  each  planet,  similar  to  what  we  have  al- 
ready stated  with  respect  to  their  inclinations,  viz.  that  if  the  mam  of 
each  planet  he  multiplied  hy  the  square  root  of  the  axis  of  its  orbit,  and 
the  product  hy  the  square  of  its  excentricityf  the  sum  ofa>ll  such  products 
throughout  the  system  is  invariable;  and  as,  in  point  of  fact,  this  sum  is 
extremely  small,  so  it  will  always  remain.  Now,  since  the  axes  of  the 
orbits  are  liable  to  no  secular  changes,  thb  is  equivalent  to  saying  that  no 
one  orbit  shall  increase  its  excentrioity,  unless  at  the  expense  of  a  common 
fund,  the  whole  amount  of  which  is,  and  must  for  ever  remain,  extremely 
minute.*  ..»,,.■,  ...   .  ,. ^. • 

*  There  is  nothing  in  this  relation,  however,  taken  per  «e,  to  secure  the  smaller  pla- 
nets — Mercury,  Mars,  Juno,  Ceres,  &c. — from  a  catastrophe,  could  they  accumulate 
on  themselves,  or  any  one  of  them,  the  whole  amount  of  this  eMcentrkily  fund.  But 
that  can  never  be :  Jupiter  and  Saturn  will  always  retain  the  lion's  share  of  it.  A 
similar  remark  applies  to  the  inelinatianfund  of  art.  639.  These  fund$,  be  it  observed, 
can  never  get  into  debt.    Every  term  of  them  is  essentially  poative.  , 


Ik' 


COMPOUND  MOTION  OF  THE  UPPER  FOCUS. 


887 


^Vfa>' 


.'  T.  CHAPTER  XIV.  ''Z  "'.'''l'"f  :V";'^! 


OP  THE  INEQUALITIES  INDEPENDENT  OP  THE  EXCENTRIOITIES. — THK 
moon's  VARIATION  AND  PARALLACTIC  INEQUALITY.  —  ANALOGOUS 
PLANETARY  INEQUALITIES. — THREE  CASES  OF  PLANETARY  PERTUR- 
BATION DISTINGUISHED.  —  OP  INEQUALITIES  DEPENDENT  ON  THE 
EXCENTRICITIES. — LONG  INEQUALITY  OP  JUPITER  AND  SATURN. — 
LAW  OP  RECIPROCITY  BETWEEN  THE  PERIODICAL  VARIATIONS  OP 
THE  ELEMENTS  OP  BOTH  PLANETS. — ^LONO  INEQUALITY  OP  THE  EARTH 
AND  VENUS. — VARIATION  OP  THE  EPOCH. — INEQUALITIES  INCIDENT 
ON  THE  EPOCH  APPECTING  THE  MEAN  MOTION. — INTERPRETATION  OP 
THE  CONSTANT  PART  OP  THESE  INEQUALITIES. — ANNUAL  EQUATION 
OP  THE  MOON.  —  HER  SECULAR  ACCELERATION. — LUNAR  INEQUALI- 
TIES DUE  TO  THE  ACTION  OP  VENUS. — EFFECT  OP  THE  SPHEROIDAL 
FIGURE  OP  THE  EARTH  AND  OTHER  PLANETS  ON  THE  MOTIONS  OF 
THEIR  SATELLITES. — OP  THE  TIDES. — MASSES  OF  DISTURBING  BODIES 
DEDUCTBLE  PROM  THE  PERTURBATIONS  THEY  PRODUCE.  —  MASS  OP 
THE  MOON,  AND  OP  JUPITER'S  SATELLITES,  HOW  ASCERTAINED. — 
PERTURBATIONS  OP  URANUS  RESULTIPG  IN  THE  DISCOVERY  OP 
NEPTUNE.  .       . 


(702.)  To  calculate  the  actual  place  of  a  planet  or  the  moon,  in  longi- 
tude and  latitude  at  any  assigned  time,  it  is  no'^  enough  to  know  the 
changes  produced  by  perturbation  in  the  elements  of  its  orbit,  still  less  to 
know  the  secular  changes  so  produced,  which  arj  only  the  outstanding  or 
uncompensated  portions  of  much  greater  chauges  induced  in  short  periods 
of  configuration.  We  must  be  enabled  to  estimate  the  actual  effect  on  its 
longitude  of  those  periodical  accelerations  and  retardations  in  the  rate  of 
its  mean  angular  motion,  and  on  its  latitude  of  those  deviations  above  and 
below  the  mean  plane  of  its  orbit,  which  result  from  the  continued  actiou 
of  the  perturbative  forces,  not  as  compensated  in  long  periods,  but  as  in 
the  act  of  their  generation  and  destruction  in  short  ones.  In  this  chapter 
we  purpose  to  give  an  account  of  some  of  the  most  prominent  of  the 
eguatiohs  or  inequalities  thence  arising,  several  of  which  are  of  high  his- 
torical interest,  as  having  become  known  by  observation  previous  to  the 


"''  n 


888 


OUTLINES   OF  ASTRONOMY. 


discovery  of  their  theoretical  causes,  and  as  having,  by  their  successive 
explanations  from  the  theory  of  gravitation,  removed  trhat  were  in  some 
instances  regarded  as  formidable  objections  against  that  theory,  and  afforded 
in  all  most  satisfactory  and  triumphant  verifications  of  it. 

(703.)  We  shall  begin  with  those  which  compensate  themselves  in  a 
synodic  revolution  of  the  dbturbed  and  disturbing  body,  and  which  are 
independent  of  any  permanent  excentricity  of  either  orbit,  going  through 
their  changes  and  effecting  their  compensation  in  orbits  slightly  elliptic, 
almost  precisely  as  if  they  were  circular.  These  inequalities  result,  in 
fact,  from  a  circulation  of  the  true  upper  focus  of  the  disturbed  ellipse 
about  its  mean  place  in  a  curve  whose  form  and  magnitude  the  principles 
laid  down  in  the  last  chapter  enable  us  to  assign  in  any  proposed  case. 
If  the  disturbed  orbit  be  circular,  this  mean  place  coincides  with  its  cen- 
tre :  if  elliptic,  with  the  situation  of  its  upper  focus,  as  determined  from 
the  principles  laid  down  in  the  last  chapter. 

(704.)  To  understand  the  nature  of  this  circulation,  we  must  consider 
the  joint  action  of  the  two  elements  of  the  disturbing  force.  Suppose  II 
to  be  the  place  of  the  upper  focus,  corresponding  to  any  situation  P  of  the 

Fig.  95. 


i 


disturbed  body,  and  let  P  P'  be  an  infinitesimal  element  of  its  orbit,  de- 
scribed in  an  instant  of  time.  Then  supposing  no  disturbing  force  to  act, 
P  F  will  be  a  portion  of  an  ellipse,  having  H  for  its  focus,  equally 
whether  the  point  P  or  F  be  regarded.  But  now  let  the  disturbing 
forces  act  during  the  instant  of  describing  P  P'.  Then  the  focus  H  will 
shift  its  position  to  H'  to  find  which  point  we  must  recollect,  1st.  What  is 
demonstrated  (in  art.  671),  viz.  that  the  effect  of  the  normal  force  is  to 
vary  the  position  of  the  line  F  H  so  as  to  make  the  angle  H  P  H'  equal 
to  double  the  variation  of  the  angle  of  tangency  due  to  the  action  of  that 


"variation"  op  the  moon  explained. 


889 


force,  Mrithout  altering  the  distance  P  H :  so  that  in  virtue  of  the  normal 
force  alone,  H  would  move  to  a  poiui.  h,  along  the  line  H  Q,  drawn  from 
H  to  a  point  Q,  90**  in  advance  of  P,  (because  S  H  being  exceedingly 
tmall,  the  angle  P  H  Q  may  be  taken  as  a  right  angle  when  P  S  Q  is  so,) 
H  approaching  Q  if  the  normal  force  act  outwards,  but  receding  from  Q 
if  inwards.  And  similarly  the  effect  of  the  tangential  force  (art.  670)  is 
to  vary  the  position  of  H  in  the  direction  H  P  or  P  H,  according  as  the 
force  retards  or  accelerates  P's  motion.  To  find  H'  then  from  H  draw 
H  P,  H  Q,  to  P  and  to  a  point  of  Fs  orbit  90"  in  advance  of  P.  On 
H  Q  take  H  A,  the  motion  of  the  focus  due  to  the  normal  force,  and  on 
H  P  take  H  k  the  motion  due  to  the  tangential  force ;  complete  the 
parallelogram  H  H',  and  its  diagonal  H  H'  will  be  the  element  of  the 
true  path  of  H  in  virtue  of  the  joint  action  of  both  forces. 

(705.)  The  most  conspicuous  case  in  the  planetary  system  to  which  the 
above  reasoning  is  applicable,  is  that  of  the  moon  disturbed  by  the  sun. 
The  inequality  thus  arising  is  known  by  the  name  of  the  moon's  varia- 
tion, and  was  discovered  so  early  as  about  the  year  975  by  the  Arabian 
astronomer  Aboul  Wefa.'  Its  magnitude  (or  the  extent  of  fluctuation  to 
and  fro  in  the  moon's  longitude  which  it  produces)  is  considerable,  being 
no  less  than  1°  4',  and  it  is  otherwise  interesting  as  being  the  first  ine- 
quality produced  by  perturbation,  which  Newton  succeeded  in  explaining 
by  the  theory  of  gravity.  A  good  general  idea  of  its  nature  may  be 
formed  by  considering  the  direct  action  of  the  disturbing  forces  on  the  moon, 
supposed  to  move  in  a  circular  orbit.  In  such  an  orbit  undisturbed,  the 
velocity  would  be  uniform ;  but  the  tangential  force  acting  to  accelerate 
her  motion  through  the  quadrants  preceding  her  conjunction  and  oppo- 
sition, and  to  retard  it  through  the  alternate  quadrants,  it  ip  evident  that 
the  velocity  will  have  two  maxima  and  two  minima,  the  former  at  the 
syzygies,  the  latter  at  the  quadratures.  Hence  at  the  syzygies  the  velocity 
will  exceed  that  which  corresponds  to  a  circular  orbit,  and  at  quadratures 
will  fall  short  of  it.  The  true  orbit  will  therefore  be  less  curved  or  more 
flattened  than  a  circle  in  syzygies,  and  more  curved  (i.  e.  protuberant  be- 
yond a  circle)  in  quadratures.  This  would  be  the  case  even  were  the 
normal  force  not  to  act.  But  the  action  of  that  force  increases  the  effect 
in  question,  for  at  the  syzygies,  and  as  far  as  64**  14'  on  either  side  of 
them,  it  acts  outwards,  or  in  counteraction  of  the  earth's  attraction,  and 
thereby  prevents  the  orbit  from  being  so  much  curved  as  it  otherwise 
would  be ;  while  at  quadratures,  and  for  25°  46'  on  either  side  of  them, 
it  acts  inwards,  aiding  the  earth's  attraction,  and  rendering  that  portion 

'Sedillot,  Nouvelles  Recherches  pour  servir  a  I'Histoire  de  rAstronomie  chez  .es 
Arabea. 


800 


OUTLINES  OF  ASTRONOMY. 


of  the  orbit  more  curved  than  it  otherwise  woald  be.  Thus  the  joint 
action  of  both  forces  distorts  the  orbit  from  a  circle  into  a  flattened  or 
elliptic  form,  having  the  longer  axis  in  quadratures,  and  the  shorter  in 
syzygies ;  and  in  this  orbit  the  moon  moves  faster  than  with  her  mean 
velocity  at  syzygy  (i.  e.  where  she  is  nearest  the  earth)  and  slower  at 
quadratures  where  farthest.  Her  angular  motion  about  the  earth  is  there- 
fore for  both  reasons  greater  in  the  former  than  in  the  latter  situation. 
Hence  at  syzygy  her  true  longitude  seen  from  the  earth  will  be  in  the  act 
of  gaining  on  her  mean, — in  quadratures  of  losing,  and  at  some  interme- 
diate points  (not  very  remote  from  the  octants)  will  neither  be  gaining 
nor  losing.  But  at  these  points,  having  been  gaming  or  losing  through 
the  whole  previous  90°  the  amount  of  gain  or  loss  will  have  attained  its 
maximum.  Consequently  at  the  octants  the  true  longitude  will  deviate 
most  from  the  mean  in  excess  and  defect,  and  the  inequality  in  question 
is  therefore  nil  at  syz^gies  and  quadratures,  and  attains  its  maxima  in 
advance  or  retardation  at  the  octants,  which  is  agreeable  to  observation. 

(706.)  Let  us,  however,  now  see  what  account  can  be  rendered  pf  this 
inequality  by  the  simultaneous  variations  of  the  axis  and  excentricity  as 
above  explained.  The  tangential  force,  as  will  be  recollected,  is  nil  at 
syzygies  and  quadratures,  and  a  maximum  at  the  octants,  accelerative  in 
the  quadrants  E  A  and  D  B,  and  retarding  in  A  D  and  B  E.  In  the  two 
former  then  the  axis  is  in  process  of  lengthening ;  in  the  two  latter,  short- 
ening. On  the  other  hand  the  normal  force  vanishes  at  (a,  &,  d,  e)  64° 
14'  from  the  syzygies.  It  acts  outwards  over  e  A  a,  i  B  c^,  and  inwards 
over  aDb  and  <£ E e.  In  virtue  of  the  tangential  force,  then,  the  point 
H  moves  towards  P  when  P  is  in  A  D,  B  E,  and  from  it  when  in  D  B, 
E  A,  the  motion  being  nil  when  at  A,  B,  D,  E,  and  most  rapid  when  at 
the  octant  D,  at  which  points,  therefore,  (so  far  as  this  force  is  concerned,) 
the  focus  H  would  have  its  mean  situation.  And  in  virtue  of  the  normal 
focus,  the  motion  of  H  in  the  direction  H  Q  will  be  at  its  maximum  of 
rapidity  towards  Q  at  A,  or  B,  from  Q  at  D  or  E,  and  nil,  at  a,  h,  d,  e.  It 
will  assist  us  in  following  out  these  indications  to  obtain  a  notion  of  the 
form  of  the  curve  really  described  by  H,  if  we  trace  separately  the  paths 
which  H  would  pursue  in  virtue  of  either  motion  separately,  since  its  true 
motion  will  necessarily  result  from  the  superposition  of  these  partial  mo- 
tions, because  at  every  instant  they  are  at  right  angles  to  each  other,  and 
therefore  cannot  interfere.  First,  then,  it  is  evident,  from  what  we  have 
fiaid  of  the  tangential  force,  that  when  P  is  at  A,  II  is  for  an  instant  at 
rest,  but  that  as  P  removes  from  A  towards  D,  H  continually  approaches 
P  along  their  line  of  junction  H  P,  which  is,  therefore,  at  each  instant  a 
tangent  to  the  path  of  H.     When  P  is  in  the  octant^  H  is  at  its  mean 


'■-■--{-■'---y^—-'- 


"variatiom"  of  the  moon  explained. 


891 


distance  from  P  (equal  to  P  S),  and  is  then  in  the  act  of  approaching  P 
most  rapidly.  From  thence  to  the  quadrature  D  the  movement  of  H  to- 
wards P  decreases  in  rapidity  till  the  quadrature  is  attained,  when  H  rests 
for  an  instant,  and  then  begins  to  reverse  its  motion,  and  travel  from  P 
at  the  same  rate  of  progress  as  before  towards  it.  Thus  it  is  clear  that, 
in  virtue  of  the  tangential  force  alone,  H  would  describe  a  four-cusped 
curve  a,d,  b,  e,  its  direction  of  motion  round  S  in  this  curve  being  oppo* 
site  to  that  of  P,  so  that  A  and  a,  D  and  d,  B  and  b,  IS  and  e,  shall  be 
corresponding  points. 

(707.)  Next  as  regards  the  normal  force.  When  the  moon  is  at  A  the 
motion  of  H  is  tcwards  D,  and  is  at  its  maximum  of  rapidity,  but  slackens 
as  P  proceeds  towards  D  and  as  Q  proceeds  towards  B.  To  the  curve 
described,  H  Q  will  be  always  a  tangent,  and  since  at  the  neutral  point  of 
the  normal  force  (or  when  P  is  64°  14'  from  A,  and  Q  64°  14'  from  D), 
the  motion  of  H  is  for  an  instant  nil  and  is  then  reversed,  the  curve  will 
have  a  cusp  at  I  corresponding,  and  H  will  then  begin  to  travel  along  the 
arc  I  m,  while  P  describes  the  corresponding  arc  from  neutral  point  to 
neutral  point  through  D.  Arrived  at  the  neutral  point  between  D  and  B, 
the  motion  of  II  along  Q  H  will  be  again  arrested  and  reversed,  giving 
rise  to  another  c\jsp  at  m,  and  so  on  Thus,  in  pirliie  of  the  normal  force 
acting  alone,  the  path  of  H  woui  i  be  the  four-cusped,  elongated  curve 
Imno,  described  with  a  motion  round  S  the  reverse  of  P's,  and  having 
a,  d,  h,  e  for  points  corresponding  to  A,  B,  D,  E,  places  of  P. 

(708.)  Nothing  is  now  easier  than  to  superpose  these  motions.     Sup- 


892 


;?>      OUTLINES   OF  ASTRONOMY. 


,ft,:T-'*  Fig.  97.      «'J^4  ■^■i^i' 


>\v^ 


\        i-i 


posing  H|,  Hg  to  be  the  points  in  either  curve  corresponding  to  P,  We* 
have  nothing  to  do  but  to  set  from  off  8,Sh  equal  and  parallel  to  S  H^ 
in  the  one  curve  and  from  h,hl3.  equal  and  parallel  to  S  Hg  in  the  other. 
Let  this  be  done  for  every  corresponding  point  in  the  two  curves,  and 
there  results  an  oval  cnneabde,  having  for  its  semiaxis  Sa=sSai-f  So,; 
and  ScfssSe^i  +  Scfj.  And  this  will  be  the  true  path  of  the  upper  focus, 
the  points  a,  d,  ft,  c,  corresponding  to  A,  D,  B,  E,  places  of  P.  And  from 
this  it  follows,  1st,  that  at  A,  B,  the  sjzygies,  the  moon  is  in  perigee  in 
her  momentary  ellipse,  the  lower  focus  being  nearer  than  the  upper. 
2dly,  That  in  quadratures  D,  E,  the  moon  is  in  apogee  in  her  then  mo- 
mentary ellipse,  the  upper  focus  being  then  nearer  than  the  lower.    3dly, 


^        e: 


y  X H 


That  H  revolves  in  the  oval  a  cf  6  e  the  contrary  way  to  P  in  its  orbit, 
making  a  complete  revolution  from  syzygy  to  syzygy  in  one  synodic  revo- 
lution of  the  moon.  ,     j  •  </        _  .       , 
(709.)  Taking  1  for  the  moon's  mean  distance  from  the  earth,  suppose 


"VARIATIOK"  OF  THE  MOON  EXPLAINED. 


898 


we  represent  Sa,  or  Stf,  (for  they  are  equal)  by  2a,  Sa,  by  26,  and  Bd,  by 
2c,  then  will  the  semiaxes  of  the  oval  adbCfSa  and  Sd  be  respectiyely 
2a+2b  and  2a+2c,  so  that  the  ezcentricitics  of  the  momentary  ellipses 
at  A  and  D  will  be  respectively  a+b  and  a-\-c.  The  total  amount  of  the 
effect  of  the  tangential  force  on  the  axisy  in  passing  from  syzygy  to  qua- 
rature,  will  evidently  be  equal  to  the  length  of  the  curvilinear  arc  a,  di 
(Jig.  art.  708),  which  is  necessarily  less  than  Soi+Sc^i  or  4a.  There- 
fore the  total  effect  on  the  aemiaxis  or  distance  of  the  moon  is  less  than 
2a,  and  the  excess  and  defect  of  the  greatest  and  least  values  of  this  dis< 
tance  thus  varied  above  and  below  the  mean  value  S  A  =  1  (which  call  a) 
will  be  less  than  a.  The  moon  then  is  moving  at  A  in  the  joerii/ee  of  an 
ellipse  whose  semiaxis  is  1+a  and  excentricity  a-{-b,  so  that  its  actual 
distance  from  the  earth  there  is  1+a — a  —  b,  which  (because  a  is  less 
than  a)  is  less  than  1  —  b.  Again,  at  D  it  is  moving  in  apogie  of  an 
ellipse  whose  semiaxis  is  1 — a  and  excentricity  a+c,  so  that  its  distance 
then  from  the  earth  is  1  — a+a+c,  which  (a  being  greater  than  a)  is 
greater  than  1+c,  the  latter  distance  exceeding  the  former  by  2a  —  2a + 
6+c. 

(710.)  Let  us  next  consider  the  corresponding  changes  induced  upon 
the  angular  velocity.  Now  it  is  a  law  of  elliptic  motion  that  at  different 
points  ^f  different  ellipses,  each  differing  very  little  from  a  circle,  the  an- 
gular velocities  are  to  each  other  as  the  square  roots  of  the  semiaxes 
directly,  and  as  the  squares  of  the  distances  inversely.  In  this  case  the 
semiaxes  at  A  and  D  are  to  each  other  as  1 +o  to  1  — a,  or  as  1  : 1  —  2a, 
BO  that  their  square  roots  are  to  each  other  as  1  : 1  —  a.  Again,  the  dis- 
tances being  to  each  other  as  l-|-o — a  —  6:1 — a+a+c,  the  inverse 
ratio  of  their  squares  (since  a,  a,  b,  c,  are  all  very  small  quantities)  is  that 
of  1— 2o-i-2a+2c  :  l-|-2o— 2a— 26,cra8l :  1— 4o— 4a— 26— 2c. 
The  angular  velocities  then  are  to  each  other  in  a  ratio  compounded  of  these 
two  proportions,  that  is  in  the  ratio  of  . 

1  :  l+3a  — 4a— 26  — 2c, 
which  is  evidently  that  of  a  greater  to  a  less  quantity.     It  is  obvious  also^ 
from  the  constitution  of  the  second  term  of  this  ratio,  that  the  normal 
force  is  far  more  influential  in  producing  this  result  than  the  tangential. 

(711.)  In  the  foregoing  reasoning  the  sun  has  been  regarded  as  fixed. 
Let  us  now  suppose  it  in  motion  (in  a  circular  orbit),  then  it  is  evident  that 
at  equal  angles  of  elongation  (of  P  from  M  seen  from  S),  equal  disturb- 
ing forces,  both  tangential  and  normal,  will  act :  only  the  syzygies  and 
quadratures,  as  well  as  the  neutral  points  of  the  normal  ^brce,  instead  of 
being  points  fixed  in  longitude  on  the  orbit  of  the  moon,  will  advance  on 
that  orbit  with  a  uniform  angular  motion  equal  to  the  angular  motion  of 


894 


ni 


OUTLINES   OP  ASTRONOMY.   lilAV  *♦ 


the  sun.  The  cuspidated  curves  n,  r/,  A, «,  and  a,  r7, 6,  e„  fig.  art.  708,  will, 
thorcfure,  no  longer  be  re-entering  curves ;  but  each  will  have  its  cusps 
screwed  round  as  it  were  in  the  direction  of  the  sun's  motion,  so  as  to  in- 
crease the  angles  between  them  in  the  ratio  of  the  tynodical  to  the  side- 
real revolution  of  the  moon  (art.  418).  And  if,  in  like  manner,  the  mo- 
tions in  these  two  curves,  thus  separately  described  by  H,  be  compounded, 
the  resulting  curve,  though  still  (loosely  speaking)  a  species  of  oval,  will 
not  return  into  itself,  but  will  make  successive  spiroidal  convolutions  about 
S,  its  farthest  and  nearest  points  being  in  the  same  ratio  more  than  90° 
asunder.  And  to  this  movement  that  of  the  moon  herself  will  conform, 
describing  a  species  of  elliptic  spiroid,  having  its  least  distances  always  in 
the  line  of  syzygies  and  its  greatest  in  that  of  quadratures.  It  is  evident, 
also,  that,  owing  to  the  longer  continued  action  of  both  forces,  t.  e.  owing 
to  the  greater  aro  over  which  their  intensities  increase  and  decrease  by 
equal  steps,  the  branches  of  each  curve  between  the  cusps  will  be  longer, 
and  the  cusps  themselves  will  be  more  remote  from  S,  and  in  the  same 
degree  will  the  dimensions  of  the  resulting  oval  be  enlarged,  and  ivritb 
them  the  amount  of  the  inequality  in  the  moon's  motion  which  they 
represent. 

(712.)  In  the  above  reasoning  the  sun's  distance  is  supposed  so  great, 
that  the  disturbing  forces  in  the  semi-orbit  nearer  to  it  shall  not  sensibly 
differ  from  those  in  the  more  remote.  The  sun,  however,  is  actually 
nearer  to  the  moon  in  conjunction  than  in  opposition  by  about  one  two- 
hundredth  part  of  its  whole  distance,  and  this  suffices  to  give  rise  to  a 
very  sensible  inequality  (called  the  parallactic  inequality)  in  the  lunar 
motions,  amounting  to  about  2'  in  its  effect  on  the  moon's  longitude,  and 
having  for  its  period  one  synodical  revolution  or  one  lunation.  As  this 
inequality,  though  subordinate  in  the  case  of  the  moon  to  the  great  ine- 
quality of  the  variation  with  which  it  stands  in  connexion,  becomes  a 
prominent  feature  in  the  system  of  inequalities  corresponding  to  it  in  the 
planetary  perturbations  (by  reason  of  the  very  great  variations  of  their 
distances  from  conjunction  to  opposition,)  it  will  be  necessary  to  indicate 
what  modifications  this  consideration  will  introduce  into  the  forms  of  our 
focus  curves,  and  of  their  superposed  oval.  Recurring  then  to  our  figures 
In  art.  706,  707,  and  supposing  the  moon  to  set  out  from  E,  and  the 
upper  focus,  in  each  curve  from  e,  it  is  evident  that  the  intercuspidal  arcs 
ea,ad,  in  the  one,  and  ep, pal,  Id,  in  the  other,  being  described  under 
the  influence  of  more  powerful  forces,  will  be  greater  than  the  arcs  d  b, 
b e,  and  dm,  mhn^ne  corresponding  in  the  other  half  revolution.  The 
two  extremities  of  these  curves  then,  the  initial  and  terminal  places  of  c 
in  each,  will  not  meet,  and  the  same  conclusion  will  hold  respecting  those 


**yARIATION"  or  THE  MOON  EXPLAINED. 


895 


ttt-  ■'■,■■}  ~<v;  «,  ♦■. 

1        .     *         Fig.  99. 

it* 

■i   > 

.  ;    ■■■  it<    ■.-. 

.      . 

'.»<j.'ii-i\ 

llilf      :^.  .,-  i!j     .1     1  • 
1 

«    •^v^     .,<  ^  . 

■rtf»« 

1-               ^     1    ;   J          I 

'              1 

^  '-.'•  1 1  ^'^"^^^y^ 

>        ■*           .'. 

'  * 

» 

1   i    »    ,  '  '        ■  '<        w  J       r-t    ,«   .  1,   '    • 

h 

'«. 

.;,;!' 

^-    .,.,...  ■■!■    V  . 

%-                                   •     ,             .:      •      ..    -^^ 

.  : 

-  ..    ■■  1     r 

•    y 


of  the  compound  oval  in  vhich  the  focus  really/  revolves,  which  will, 
therefore,  be  as  in  the  annexed  figure.  Thus,  at  the  end  of  a  complete 
lunation,  the  focus  will  have  shifted  its  place  from  e  to /in  a  line  parallel 
to  the  line  of  quadratures.  The  next  revolution,  and  the  next,  the  same 
thing  would  happen.  Meanwhile,  however,  the  sun  has  advanced  in  its 
orbit,  and  the  line  of  quadratures  has  changed  its  situation  by  an  equal 
angular  motion.  In  consequence,  the  next  terminal  situation  (g)  of  the 
forces  will  not  lie  in  the  line  e/  prolonged,  but  in  a  line  parallel  to  the 
new  situation  of  the  line  of  quadratures,  and  this  process  continuing,  will 
evidently  give  rise  to  a  movement  of  circulation  of  the  point  e,  round  a 
mean  situation  in  an  annual  period ;  and  this,  it  is  evident,  is  equivalent 
to  an  annual  circulation  of  the  central  point  of  the  compound  oval  itself, 
in  a  small  orbit  about  its  mean  position  S.  Thus  we  see  that  no  perma- 
nent  and  indefinite  increase  of  excentricity  can  arise  from  this  cause; 
which  would  be  the  case,  however,  but  for  the  annual  motion  of  the  sun. 
(713.)  Inequalities  precisely  similar  in  principle  to  the  variation  and 
parallactic  inequality  of  the  moon,  though  greatly  modified  by  the  different 
relations  of  the  dimensions  of  the  orbits,  prevail  in  all  cases  where  planet 
disturbs  planet.  To  what  extent  this  modification  is  carried  will  be  evi- 
dent, if  we  cast  our  eyes  on  the  examples  given  in  art.  612,  where  it  will 
be  seen  that  the  disturbing  force  in  conjunction  often  exceeds  that  in 
opposition  in  a  very  high  ratio,  (being  in  the  case  of  Neptune  disturbing 
Uranus  more  than  ten  times  as  great.)  The  effect  will  be,  that  the  orbit 
described  by  the  centre  of  the  compound  oval  about  S,  will  be  much 
greater  relatively  to  the  dimensious  of  that  oval  itself,  than  in  the  case  of 
the  moon.  Bearing  in  mind  the  nature  and  import  of  this  modification, 
we  may  proceed  to  inquire,  apart  from  it,  into  the  number  and  distribution 
of  the  undulations  in  the  contour  of  the  oval  itself,  arising  from  tho  alter* 


896 


'v    OUTLINES  OP  ASTRONOMY.      y.AV    ' 


nations  of  direction /)/u«  and  mintu  <>f  the  disturbing  forces  in  the  course 
of  a  synodic  revolution.  But  flrst  it  should  be  mentioned  that,  in  the 
case  of  an  exterior  disturbed  by  an  interior  plauct,  the  distui^liig  body's 
angular  motion  exceeds  that  of  the  disturbed.  Hence  P.  i^iiough  advan- 
cing in  its  orbit,  recedes  relatively  to  the  line  of  syzygie%,  or,  which  comes 
to  the  same  thing,  the  neutral  points  of  either  force  overtake  it  in  succcs- 
sion,  and  each,  as  it  comes  up  to  it,  gives  rise  to  a  cusp  in  the  corresponding 
/ocus  curve.  The  angles  between  the  successive  cusps  will  therefore  bo 
to  the  angles  between  the  corresponding  neutral  points  for  a  fixed  position 
of  M,  in  the  same  constant  ratio  of  the  synodic  to  the  sidereal  period  of  V, 
which  however  is  now  a  ratio  of  less  inequality.  These  angles  thcr.  w.'^ 
be  contracted  in  amplitude,  and,  for  the  same  reason  as  before,  tlic  cxcun 
sions  of  the  focus  will  be  diminished,  and  the  more  so  the  snorter  tiiv. 
synodic  revolution. 

(714.)  Since  the  cusps  of  either  curve  recur,  in  success  ve  bynodic 
revolutions  in  the  same  order,  and  at  the  same  angular  diEitancce  from 
each  other,  and  from  the  line  of  conjunction,  the  a\me  will  be  true  of  1  all 
the  corresponding  points  in  the  curve  resulting  from  their  suporposiition. 
In  that  curve,  every  cusp,  of  either  constituent,  will  give  rise  to  a  con- 
vexity, and  every  intercuspidal  arc  to  a  relative  concavity.  It  is  evident 
then  that  the  compound  curve  or  true  path  of  the  focus  so  resulting,  but 
for  the  cause  above  mentioned,  would  return  into  itself,  whenever  the 
periodic  times  of  t! '  jisturbing  and  disturbed  bodies  are  commensurate, 
because  in  thn,t  case  the  synodic  period  will  also  be  commensurate 
with  either,  aad  the  tire  of  longitude  intercepted  between  the  sidereal 

Fig.  100. 


place  of  any  one  conjunction,  and  that  next  following  it,  will  be  an  ali- 
quot part  of  360°.  In  all  other  cases  it  would  be  a  non-reentering,  more  or 
less  nudulating  and  more  or  less  regular,  spiroid,  according  to  the  number 
'»''  ou:p8  in  t  ich  of  the  constituent  curves  (that  is  to  say,  according  to 
^^e  numbei  of  neutral  points  or  changes  of  direction  i  from  inwards  to 


ANALOGOUS   PLANETARY  DISTURBANCES. 


897 


outwuriln,  or  from  a(H:cloriitiiig  to  retarding,  auJ  vice  veraa,  of  the  normal 
utid  tangoDtiuI  foroei, >  in  u  complete  synodic  revolution;  and  their  distri- 
bution over  the  circuuil.  renoe. 

(716.)  With  rr  Tard  to  ih  >  ohanj:  s,  it  is  necessary  '^  distinguish 
throe  cases,  in  whieli  the  pcrturbatioui*  '^  planet  by  planet  >ro  very  dis- 
tinct in  character.  Ist.  When  the  disturbin;:  i  'met  is  exteri*  lu  this 
case  there  are  four  neutral  points  of  either  for>:c.  Those  of  tL  tangen- 
tial force  occur  at  the  syzygies,  and  i.  the  poii  3  of  the  disturbed  orbit 
(which  wo  shall  call  points  of  oquidiut  nee),  cij  lidistant  from  the  suu 
and  the  disturbing  planet  (at  which  poiu<  as  wo  have  shown  (art.  G14), 
the  total  disturbing  force  is  always  directed  inwards  towards  the  Bin.) 
Those  of  the  normal  force  occur  at  points  intermediate  between  these  1  iit 
mentioned  points,  and  the  syzygies,  which,  it  the  dinturbing  planet  be 
veri/  distant,  hold  nearly  the  situation  they  do  .u  the  lunar  theory,  i.  <:. 
considerably  nearer  the  quadratures  than  the  syzygic  .  la  proportion  as  the 
distance  of  the  disturbing  planet  diminishes,  two  of  liese  points,  viz.  those 
nearest  the  syzygy,  approach  to  each  other,  and  to  le  syzygy,  and  in  the 
extreme  case,  when  the  dimensions  of  the  orbits  are  equal;  coincide  with  it. 

(716.)  The  second  case  is  that  in  which  the  distu  -bing  planet  is  inte- 
rior to  the  disturbed,  but  at  a  distance  from  the  sun  gr  <ater  than  half  that 
of  the  latter.  In  this  case  there  are  four  neutral  poin  of  the  tangential 
force,  and  only  twc  '^f  the  normal.  Those  of  the  tan^  ntial  force  occur 
at  the  syzygies,  and  at  the  points  of  equidistance.  The  force  retards  the 
disturbed  body  from  conjunction  to  the  first  such  points  » titer  conjunction, 
accelerates  it  thence  to  the  opposition,  thence  again  retai  Is  it  to  the  next 
point  of  equidistance,  and  finally  again  accelerates  it  u^  to  the  conjunc- 
tion. As  the  disturbing  orbit  contracts  in  dimension,  the  points  of  equi- 
distance approach;  their  distance  from  syzygy  from  60^  (the  extreme 
case)  diminishing  to  nothing,  when  they  coincide  with  eaeh  other,  and 
with  the  conjunction.  In  the  case  of  Saturn  disturbed  by  Jupiter,  that 
distance  is  only  23**  33'.  The  neutral  points  of  the  normal  force  lie 
s-^mewhat  beyond  the  quadratures,  on  the  side  of  the  opposition,  and  do 
not  undergo  any  very  material  change  of  situation  with  the  contraction 
of  the  disturbing  orbit. 

(717.)  The  third  case  is  that  in  which  the  diameter  of  the  disturbing 
interior  orbit  is  lues  than  half  that  of  the  disturbed.  In  this  case  there 
are  only  two  points  of  evanescence  for  either  force.  Those  of  the  tan- 
gential force  are  the  syzygies.  The  disturbed  planet  is  accelerated  through- 
out the  whole  semi-revolution  from  conjunction  to  opposition,  and  retarded 
from  opposition  to  conjunction,  the  maxima  of  acceleration  and  retardation 
occurrmg  Uut  far  fVom  quadrature.     The  neutral  points  of  the  norm  il 


J 


398 


OUTLINES  OF  i^STRONOMY. 


force  are  situated  nearly  as  in  the  last  case ;  that  is  to  say,  beyond  the 
quadratures  towards  the  opposition.  AH  these  varieties  the  student  will 
easily  trace  out  by  simply  drawing  the  figures,  and  resolving  the  forces  in 
a  series  of  cases,  beginning  with  a  very  large  and  ending  with  a  very 
small  diameter  of  the  disturbing  orbit.  It  will  greatly  aid  him  in  im- 
pressing on  his  imagination  the  general  relations  of  the  subject,  if  be 
construct,  as  he  proceeds,  for  each  case,  the  elegant  and  symmetrical  ovals 

Fig.  101. 


in  which  the  points  N  and  L  (^fig.  art.  675,)  always  lie,  for  a  fixed  posi' 
tion  of  M,  and  of  which  the  annexed  figure  expresses  the  forms  tbey 
respectively  assume  in  the  third  case  now  under  consideration.  The 
second  only  differs  from  this,  in  having  the  common  vertex  m,  of  both 
ovals,  outside  of  the  disturbed  orbit  A  P,  while  in  the  case  of  an  exterior 
disturbing  planet,  the  oval  m  L  assunies  a  four-lobed  form ;  its  lobes 
respectively  touching  the  oval  m  N  in  its  vertices,  and  cutting  the  orbit 
A  P  in  the  points  of  equidistance  and  of  tangency,  (i.  e.  where  M  P  S  is 
a  right  angle)  as  in  this  figure. 

Fig.  102. 


THREE  CASES  DISTINQUISHED. 


399 


(718.)  It  would  be  easy  now,  bearing  these  features  in  mind,  to  trace 
in  any  proposed  case  the  form  of  the  spiroid  curve,  described,  as  above 
explained,  by  the  upper  focus.  It  will  suffice,  however,  for  our  present) 
purpose,  to  remark,  1st,  That  between  every  two  successive  conjunctions 
of  P  and  M,  the  same  general  form,  the  same  subordinate  undulations, 
and  the  same  terminal  displacement  of  the  upper  focus,  are  continually 
repeated.  2dly,  That  the  motion  of  the  focus  in  this  curve  is  retrograde 
whenever  the  disturbing  planet  is  exterior,  and  that  in  consequence  the 
apsides  of  the  momentary  ellipse  also  recede,  with  a  mean  velocity  such 
as,  but  for  that  displacement,  would  bring  them  round  at  the  each  con- 
junction to  the  same  relative  situation  with  respect  to  the  line  jf  syzygies. 
3dly,  That  in  consequence  of  this  retrograde  movement  of  -  iie  apse,  the 
disturbed  planet,  apart  from  that  consideration,  would  be  t  ^ice  in  peri- 
bclio  and  twice  in  aphelio  in  its  momentary  ellipse  in  each  synodic  revo- 
lution, just  as  in  the  case  of  the  moon  disturbed  by  the  sur. — and  that  in 
consequence  of  this  and  of  the  undulating  movement  of  the  focus  H  it- 
self, an  inequality  will  arise,  analogous,  mutatis  mutandis  in  each  case,  to 
the  moon's  variation ;  under  which  term  we  comprehend  (not  exactly  in 
conformity  to  its  strict  technical  meaning  in  the  lunar  theory)  not  only 
the  principal  inequality  thus  arising,  but  all  its  subordinate  fluctuations. 
And  on  this  the  parallactic  inequality  thus  violently  exaggerated  is 
superposed. 

(719.)  Wo  come  now  to  the  class  of  inequalities  which  depend  for 
their  existence  on  an  appreciable  amount  of  permanent  excentricity  in 
the  orbit  of  one  or  of  both  the  disturbing  and  disturbed  planets,  in  con- 
sequence of  which  all  their  conjunctions  do  not  take  place  at  equal 
distances  either  from  the  central  body  or  from  each  other,  and  therefore 
that  symmetry  in  every  synodic  revolution  on  which  depends  the  exact 
restoration  of  both  the  axis  and  excentricity  to  their  original  values  at  the 
completion  of  each  such  revolution  no  longer  subsists.  In  passing  from 
conjunction  to  conjunction,  then,  there  will  no  longer  be  effected  a  com- 
plete restoration  of  the  upper  focus  to  the  same  relative  situation,  or  of 
the  axis  to  the  same  length,  which  they  respectively  had  at  the  outset. 
At  the  same  time  it  is  not  less  evident  that  the  differences  in  both  re- 
spects are  only  what  remain  outstanding,  after  the  compensation  of  by  far 
the  greater  part  of  the  deviations  to  and  fro  from  a  mean  state,  which 
occur  in  the  course  of  the  revolution ;  and  that  they  amount  to  but  small 
fractions  of  the  total  excursions  of  the  focus  from  its  first  position,  or  of 
the  increase  and  decrease  in  the  length  of  the  axis  effected  by  the  direct 
action  of  the  tangential  force,  —  so  small,  indeed,  that,  unless  owing  to 
peculiar  adjustments  they  be  enabled  to  accumulate  again  and  again  at 


'   \K 


iSitf: 


%       .V 


400 


OUTLINES  OF  ASTRONOMY.'^ 


Buocessive  oonjunctions  in  the  same  direction,  they  would  be  altogether 
undeserving  of  any  especial  notice  in  a  work  of  this  nature.  Such  ad- 
justments, however,  would  evidently  exist  if  the  periodic  times  of  the 
planets  were  exactly  commensurable;  since  in  that  case  all  the  possible 
oonjunctions  which  could  ever  happen  (the  elements  not  being  materially/ 
changed)  would  take  place  at  fixed  points  in  longitude,  the  intermediate 
points  being  never  visited  by  a  conjunction.  Now,  of  the  conjunctions 
thus  distributed,  their  relations  to  the  lines  of  symmetry  in  the  orbits 
being  all  dissimilar,  some  one  must  be  more  influential  than  the  rest  on 
each  of  t\e  elements  (not  necessarily  the  same  upon  all).  Consequently, 
in  a  complete  cycle  of  conjunctions,  wherein  each  has  been  visited  in  its 
turn,  the  influence  of  that  one  on  the  element  to  which  it  stands  so  espe- 
cially related,  will  preponderate  over  the  counteracting  and  compensating 
influence  of  the  rest,  and  thus,  although  in  such  a  cycle  as  above  specified 
a  further  and  much  more  exact  compensation  will  have  been  efiected  in 
its  value  than  in  a  single  revolution,  still  that  compensation  will  not  be 
complete,  but  a  portion  of  the  effect  (be  it  to  increase  or  to  diminish  the 
excentricity  or  the  axis,  or  to  cause  the  apse  to  advance  or  to  recede,) 
will  remain  outstanding.  In  the  next  cycle  of  the  same  kind  this  will  be 
repeated,  and  the  result  will  be  of  the  same  character,  and  so  on,  till  at 
length  a  sensible  and  ultimately  a  large  amount  of  change  shall  have 
taken  place,  and  in  fact  until  the  axis  (and  with  it  the  mean  motion)  shall 
have  so  altered  as  to  destroy  the  commensurability  of  periods,  and  the  ap- 
sides have  so  shifted  as  to  alter  the  place  of  the  most  influential  conjunction. 
(720.)  Now,  although  it  is  true  that  the  mean  motions  of  no  two 
planets  are  exactly  commensurate,  yet  cases  are  not  wanting  in  which 
there  exists  an  approach  to  this  adjustment.  For  instance,  in  the  case  of 
Jupiter  and  Saturn,  a  cycle  composed  of  five  periods  of  Jupiter  and  two 
of  Saturn,  although  it  does  not  exactly  bring  about  the  same  configura- 
tion, does  so  pretty  nearly.  Five  periods  of  Jupiter  are  21663  days,  and 
two  periods  of  Saturn,  21519  days.  The  difference  is  only  146  days,  in 
which  Jupiter  describes,  on  an  average,  12®,  and  Saturn  about  5°;  so 
that  after  the  lapse  of  the  former  interval  they  will  only  be  7"  from  a 
conjunction  in  the  same  parts  of  their  orbits  as  before.  If  we  calculate 
the  time  which  will  exactly  bring  about,  on  the  average,  three  conjunc- 
tions of  the  two  planets,  we  shall  find  it  to  be  21760  days,  their  synodical 
period  being  7253-4  days.  In  this  interval  Saturn  will  have  described 
8°  6'  in  excess  of  two  sidereal  revolutions,  and  Jupiter  the  same  angle  in 
excess  of  five.  Every  third  conjunction,  then,  will  take  place  8"  6'  in 
advance  of  the  preceding,  which  is  near  enough  to  establish,  not,  it  is 
true,  an  identity  with,  but  still  a  great  approach  to  the  case  in  question. 


,h\ 


LONG  EQUATION  OF  JUPITER  AND   SATURN. 


401 


The  excess  of  action,  for  several  such  triple  conjunctions  (7  or  8)  in  suc- 
cession, will  lie  the  same  way,  and  at  each  of  them  the  elements  of  F's 
orbit  and  its  angular  motion  will  be  similarly  influenced^  so  tis  to  accu- 
mulate the  effect  upon  its  longitude ;  thus  giving  rise  to  an  irregularity 
of  considerable  magnitude  and  very  long  period,  which  is  well  known  to 
astronomers  by  the  name  of  the  great  inequality  of  Jupiter  and  Satura.'i»^f 
(721.)  The  arc  8°  §'  is  contained  44|  times  in  the  whole  circumference 
of  360°  J  and  accordingly,  if  we  trace  round  this  particular  conjunction, 
we  shall  find  it  will  return  to  the  same  point  of  the  orbit  in  so  many 
times  21760  days,  or  in  2648  years.     But  the  ;onjunction  we  are  now 
considering  is  only  one  out  of  three.     The  other  two  will  happen  at 
points  of  the  orbit  about  123°  and  246°  distant,  and  these  points  also  will 
advance  by  the  same  arc  of  8°  6'  in  21760  days.     Consequently  the 
period  of  2648  years  will  bring  them  all  round,  and  in  that  interval  each 
of  them  will  pass  through  that  point  of  the  two  orbits  from  which  we 
commenced :  hence  a  conjunction  (one  or  other  of  the  three)  will  happen 
at  that  point  once  in  one  third  of  this  period,  or  in  883  years ;  and  this 
is,  therefore,  the  cycle  in  which  the  "great  inequality"  would  undergo 
its  full  compensation,  did  the  elements  of  the  orbits  continue  all  that 
time  invariable.     Their  variation,  however,  is  considerable  in  so  long  an 
interval ;  and,  owing  to  this  cause,  the  period  itself  is  prolonged  to  about 
918  years. 

(722.),  We  have  selected  this  inequality  as  the  most  remarkable  in- 
stance of  this  kind  of  action  on  account  of  its  magnitude,  the  length  of 
its  period,  and  its  high  historical  interest.  It  had  long  been  remarked 
by  astronomers,  that  on  comparing  together  modem  with  ancient  observa- 
tions of  Jupiter  and  Saturn,  the  mean  motions  of  these  planets  did  not 
appear  to  be  uniform.  The  period  of  Saturn,  for  instance,  appeared  to 
have  been  lengthening  throughout  the  whole  of  the  seventeenth  century, 
and  that  of  Jupiter  shortening — that  is  to  say,  the  one  planet  was  con- 
stantly lagging  behind,  and  the  other  getting  in  advance  of  its  calculated 
place.  On  the  other  hand,  in  the  eighteenth  century,  a  process  precisely 
the  reverse  seemed  to  be  going  on.  It  is  true  the  whole  retardations  and 
accelerations  observed  were  not  very  great;  but,  as  their  influence  went  on 
accumulating,  they  produced,  at  length,  material  differences  between  the 
observed  and  calculated  places  of  both  these  planets,  which  as  they  could 
not  then  be  accounted  for  by  any  theory,  excited  a  high  degree  of  atten- 
tion, and  were  even,  at  one  time,  too  hastily  regarded  as  almost  subversive 
of  the  Newtonian  doctrine  of  gravity.  For  a  long  while  this  difference 
baffled  every  endeavour  to  account  for  it;  till  at  length  Laplace  pointed 
26 


402 


.•X, 


\ 


OUTLINES  OF  ASTRONOMY. 


out  its  cause  ia  the  near  commensurability  of  the  mean  motions,  as  above 
shown,  and  succeeded  in  calculating  its  period  and  amount. 

(723.)'  The  inequality  in  question  amounts,  at  its  maximum,  to  an  al- 
ternate gain  and  loss  of  about  0°  49'  in  the  longitude  of  Saturn,  and  a 
corresponding  loss  and  gain  of  about  0°  21'  in  that  of  Jupiter.  That  aa 
acceleration  in  the  one  planet  must  necessarily  be  accompanied  by  a  re- 
tardation in  the  other,  might  appear  at  first  sight  «clf-evident,  if  wo  con- 
sider, that  action  and  reaction  being  equal,  and  iu  contrary  directions, 
whatever  momentum  Jupiter  communicates  to  Saturn  in  the  direction 
P  M,  the  same  momentum  must  Saturn  communicate  to  Jupiter  in  the 
direction  M  P.  The  one,  therefore,  it  might  seem  to  be  plausibly  argued, 
will  be  dragged  forward,  whenever  the  other  is  pulled  back  in  its  orbit. 
The  inference  is  correct,  so  far  as  the  general  and  final  result  goes;  but 
the  reasoning  by  which  it  would,  on  the  first  glance,  appear  to  be  thus 
summarily  established  is  fallacious,  or  at  least  incomplete.  It  is  perfectly 
true  that  whatever  momentum  Jupiter  communicates  directly  to  Saturn, 
Saturn  communicates  an  equal  momentum  to  Jupiter  in  an  opposite  linear 
direction.  But  it  is  not  with  the  absolute'  motions  of  the  two  planets  in 
space  that  we  are  now  concerned,  but  with  the  relative  motion  of  each 
separately,  with  respect  to  the  sun  regarded  as  at  rest.  The  perturhatiuc 
forces  (the  forces  which  disturb  these  relative  motions)  do  not  act  along 
the  line  of  junction  of  the  planets  (art.  614.)  In  the  reasoning  thus 
objected  to,  the  attraction  of  each  on  the  sun  has  been  left  out  of  tho 
account',  and  it  remains  to  be  shown  that  these  attractions  neutralize  and 
destroy  each  other's  effects  in  considerable  periods  of  time,  as  bearing 
upon  the  result  in  question.  Suppose  then  that  we  for  a  moment  abandon 
the  point  of  vievy,  in  which  we  have  hitherto  all  along  considered  tho 
subject,. and  regard  the  sun  as  free  to  move,  and  liable  to  be  displaced  by 
the  attractions  of  the  two  planets.  Then  will  the  movements  of  all  be 
performed  about  the  common  centre  of  gravity,  just  as  they  would  have 
been  about  the  sun's  centre  regarded  as  immoveable,  the  sun  all  the  while 
circulating  in  a  small  orbit  (with  a  motion  compounded  of  the  two  elliptic 
motions  it  would  have  in  virtue  of  their  separate  attractions)  about  the 
same  centre.  Now  in  this  case  M  still  disturbs  P,  and  P,  M,  but  the 
whole  disturbing  force  now  acts  along  their  line  of  junction,  and  since  it 
remains  true  that  whatever  momentum  M  generates  in  P,  P  will  generate 
the  same  in  M  in  a  contrary  direction ;  it  will  also  be  strictly  true  that,  so 

'  We  &.e  here  reading  a  sort  of  recantation.  In  the  edition  of  1833  the  remarkable 
result  in  question  is  sought  to  be  established  by  this  vicious  reasoning.  The  mistake 
is  a  very  natural  one,  and  is  so  apt  to  haunt  the  ideas  of  beginners  in  this  department 
nf  physics,  that  it  is  worth  while  expressly  to  warn  them  againat,it. 


LAW  OF  RECIPROCITY. 


403 


far  as  a  disturbance  of  their  elliptic  motions  about  the  common  centre  of 
gravity  of  the  system  is  alone  regarded,  whatever  disturbance  of  velocity 
is  generated  in  the  one,  a  contrary  disturbance  of  velocity  (only  in  the 
inverse  ratio  of  the  masses  and  modified,  though  never  contradicted,  by 
the  directions  in  which  they  are  respectively  moving),  will  be  generated 
in  the  other.  Now  when  we  are  considering  only  inequalities  of  long 
period  comprehending  many  complete  revolutions  of  both  planets,  and 
which  arise  from  changes  in  the  axes  of  the  orbits,  affecting  their  mean 
motions,  it  matters  not  whether  we  suppose  these  motions  performed 
about  the  ommon  centre  of  gravity,  or  about  the  sun,  which  never  de- 
parts from  that  centre  to  any  material  extent  (the  mass  of  the  sun  being 
such  in  comparison  with  that  of  the  planets,  that  that  centre  always  lies 
within  his  surface.)  The  mean  motion  therefore,  regarded  as  the  average 
angular  velocity  during  a  revolution,  is  the  same  whether  estimated  by 
reference  to  the  sun's  centre,  or  to  the  centre  of  gravity,  or,  in  other 
words,  the  relative  mean  motion  referred  to  the  sun  is  identical  with  the 
absolute  mean  motion  referred  to  the  centre  of  gravity. 

(724.)  This  reasaning  applies  equally  to  every  case  of  mutual  disturb- 
ance resulting  in  a  long  inequality  such  as  may  arise  from  a  slow  and 
long-continued  periodical  increase  and  diminution  of  the  axes,  and  geom- 
eters have  accordingly  demonstrated  as  a  consequence  from  it,  that  the 
proportion  in  which  such  inequalities  affect  the  longitudes  of  the  two 
planets  concerned,  or  the  maxima  of  the  excesses  and  defects  of  their 
longitudes  above  and  below  their  elliptic  values,  thence  arising,  in  each, 
are  to  each  other  in  the  inverse  ratio  of  their  masses  multiplied  by  the 
square  roots  of  the  major  axes  of  their  orbits,  and  this  result  is  confirmed 
by  observation,  and  will  be  found  verified  in  the  instance  immediately  in 
question  as  nearly  as  the  uncertainty  still  subsisting  as  to  the  masses  of 
the  two  planets  will  permit.  "  •  •■  .        :'.  r 

(725.)  The  inequality  in  question,  as  has  been  observed  in  general, 
(art.  718,)  would  be  much  greater,  were  it  not  for  the  partial  compensa- 
tion which  is  operated  in  it  in  every  triple  conjunction  of  the  planets. 
Suppose  PQR  to  be  Saturn's  orbit,  and  pqr  Jupiter's;  and  suppose  a 
conjunction  to  take  place  at  V p,  on  the  line  SA;  a  second  at  123"  dis- 
tance, on  the  line  S  B;  a  third  at  246"  distance,  on  S  C  ;  and  the  next  at 
368",  on  S  D.  This  last-meutioned  conjunction,  taking  place  nearly  in 
the  situation  of  the  first,  will  produce  nearly  a  repetition  of  the  first  effect 
in  retarding  or  accelerating  the  planets ;  but  the  other  two,  being  in  the 
most  remote  situations  possible  from  the  first,  will  happen  under  entirely 
different  circumstances  as  to  the  position  of  the  perihelia  of  the  orbits. 
Now,  we  have  seen  that  a  presentation  of  the  one  planet  to  the  other  in 


404 


OUTLINBS  OF  ASTRONOMY. 


..;<^^■:  -.    iM  ■■ 


;.v  ■^"."■t:t  ■  -(:',•   -(f 


Fig.  108. 


oonjtiaction,  in  a  variety  of  situations,  tends  to  produce  compensation; 
and,  in  fact,  the  greatest  possible  amount  of  compensation  which  can  be 
pioduced  by  only  three  conjunctions  is  when  they  are  thus  equally  dis- 
tributed round  the  centre.  Hence  we  see  that  it  is  not  the  whole  amount 
of  perturbation  which  is  thus  accumulated  in  each  triple  conjunction,  bui 
only  that  small  part  which  is  left  uncompensated  by  the  intermediate 
ones.  The  reader,  who  possesses  already  some  acquaintance  with  the 
subject,  will  not  be  at  a  loss  to  perceive  how  this  consideration  is,  in  fact, 
equivalent  to  that  part  of  the  geometrical  investigation  of  this  inequality 
which  leads  us  to  seek  its  expression  in  terms  of  the  third  order,  or  in- 
volving the  cubes  and  products  of  three  dimensions  of  the  excentricities 
and  inclinations ;  and  how  the  continual  accumulation  of  small  quantities, 
during  long  periods,  corresponds  to  what  geometers  intend  when  they 
speak  of  small  terms  receiving  great  accessions  of  magnitude  by  the  intro 
duction  of  large  coefficients  in  the  process  of  integration. 

(726.)  Similar  considerations  apply  to  every  case  of  approximate  com* 
mensurability  which  can  take  place  among  the  mean  motions  of  any  two 
planets.  Such,  fcr  instance,  is  that  which  obtains  between  the  mean 
motion  of  the  earth  and  Venus, — 13  times *the  period  of  Venus  being  very 
nearly  equal  to  8  times  that  of  the  earth.  This  gives  rise  to  an  extremely 
near  coincidence  of  every  fifth  conjunction,  in  the  same  parts  of  each  orbit 
(within  slijth  part  of  a  circumfprcnce,)  and  therefore  to  a  correspondingly 
extensive  accumulation  of  the  resulting  uncompensated  perturbation. 
But,  on  the  other  hand,  the  part  of  the  perturbation  thus  accumulated  is 
only  that  which  remains  outstanding  after  passing  the  equalizing  ordeal 
of  five  conjunctions  equally  distributed  round  the  circle ;  or,  in  the  lan- 
guage of  geometers,  is  dependent  on  powers  and  products  ot  the  excen- 
tricities and  inclinations  of  the  fifth  order.  It  is,  therefore,  extremely 
minute,  and  the  whole  resulting  inequality,  according  to  the  elaborate 
•wlculations  of  Mr.  Airy,  to  whom  it  owes  its  detection,  amounts  to  no 


LONG   INEQUALITIES   OF  ELEMENTS. 


406 


more  than  a  few  seconds  at  its  maximum,  while  its  period  is  no  less  than 
240  years.  This  example  will  serve  to  show  to  what  minuteQess  these 
inquiries  have  been  carried  to  the  planetary  theory. 

(727.)  That  variations  of  long  period  arising  in  the  way  .above  described 
are  necessarily  accompanied  by  similarly  periodical  displacements  of  the 
upper  focus,  equivalent  in  their  effect  to  periodical  fluctuations  in  the 
magnitude  of  the  excentricity,  and  in  the  position  of  the  line  of  apsides, 
is  evident  from  what  has  been  already  said  respecting  the  motion  of  the 
upper  focus  under  the  influence  of  the  disturbing  forces.  In  the  case  of 
circular  orbits  the  mean  place  of  H  coincides  with  S  the  centre  of  the  sun, 
but  if  the  orbits  have  any  independent  ellipticity,  this  coincidence  will  no 
longer  exist  —  and  the  mean  place  of  the  upper  focus  will  come  to  be 
inferred  from  the  average  of  all  the  situations  which  it  actually  holds 
during  an  entire  revolution.  Now  the  fixity  of  this  point  depends  on  the 
equality  of  each  of  the  branches  of  the  cuspidated  curves,  and  consequent 
equality  of  excursion  of  the  focus  in  each  particular  direction,  in  every 
successive  situation  of  the  line  of  conjunction.  But  if  there  be  some 
one  line  of  conjunction  in  which  these  excursions  are  greater  in  any  one 
particular  direction  than  in  another,  the  mean  place  of  the  focus  will  be 
displaced,  and  if  this  process  be  repeated,  that  mean  place  will  continue 
to  deviate  more  and  more  from  its  original  position,  and  thus  will  arise  a 
circulation  of  the  mean  place  of  the  focus /or  a  revolution  about  another 
mean  situation,  the  average  of  all  the  former  mean  places  during  a  com- 
plete cycle  of  conjunctions.  Supposing  S  to  be  the  sun,  O  the  situation 
the  upper  focus  would  have,  had  these  inequalities  no  existence,  and  H  K 
the  path  of  the  upper  focus,  which  it  pursues  about  0  by  reason  of  them, 
then  it  is  evident  that  in  the  course  of  a  complete  cycle  of  the  inequality 
in  question,  the  excentricity  will  have  fluctuated  between  the  extreme 
limits  SJ  and  ST  and  the  direction  of  the  longer  axis  betweeu  the 
extreme  position  SH  and  S  K,  and  that  if  we  suppose  ijhk  to  be  the 
corresponding  mean  places  of  the  focus,  ij  will  be  the  extent  of  the  fluctu- 
ation of  the  mean  excentricity,  and  the  angle  hsk,  that  of  the  longitude 
of  the  perigee. 

(728.)  The  periods  then  in  which  these  fluctuations  go  through  their 
phases  are  necessarily  equal  in  duration  with  that  of  the  inequality  in 
longitude,  with  which  they  stand  in  connexion.  But  it  by  no  means 
follows  that  their  maxima  all  coincide.  The  variation  of  the  axis  to  which 
that  of  the  mean  motion  corresponds,  depends  on  the  tangential  force  only 
whose  maximum  is  not  at  conjunction  or  opposition,  but  at  points  remote 
from  either,  while  the  excentricity  depends  both  on  the  normal  and  tan- 
gential forces,  the  maximum  of  the  former  of  which  is  at  the  conjunction 


...  -.I'-l:, 


fc 


406 


OUTLINES  OF  ASTRONOMY. 


■^  .■    \     US!   l:.hr:  -il  n:^       Kg-  104, 


:>    •    il.i. 


That  particular  conjunction  therefore,  which  is  most  influential  on  th« 
axis,  is  not  so  on  the  excentricity,  so  that  it  can  by  no  means  be  concludc(] 
that  either  the  maximum  value  of  the  axis  coincides  with  the  maximum, 
or  the  minimum  of  the  excentricity,  or  with  the  greatest  excursion  to  or 
fro  of  the  line  of  apsides  from  its  mean  situation,  all  that  can  be  safely 
asserted  is,  that  as  either  the  axis  or  the  excentricity  of  the  one  orbit 
varies,  that  of  the  other  will  vary  in  the  opposite  direction. 

(729.)  The  primary  elements  of  the  lunar  and  planetary  orbits,  which 
may  be  regarded  as  variable,  are  the  longitude  of  the  node,  the  inclina- 
tion, the  axis,  excentricity,  longitude  of  the  perihelion,  and  epoch  (art. 
496).  In  the  foregoing  articles  we  have  shown  in  what  manner  each  of 
thr  first  five  of  these  elements  is  made  to  vary,  by  the  direct  action  of 
the  perturbing  forces.  It  remains  to  explain  in  what  manner  the  last 
comes  to  be  affected  by  them.  And  here  it  is  necessary,  in  the  first  in- 
stance, to  remove  some  degree  of  obscurity  which  may  be  thought  to  hang 
about  the  sense  in  which  the  term  itself  is  to  be  understood  in  speaking  of 
an  orbit,  every  other  element  of  which  is  regarded  as  in  a  continual  state 
of  v.-'riation.  Supposing,  then,  that  we  were  to  reverse  the  process  of 
calculation  described  in  arts.  499  and  500  by  which  a  planet's  heliocentric 
longitude  ii?  an  elliptic  orbit  is  computed  for  a  given  time  j  and  setting 
out  with  a  heMocentric  longitude  ascertained  by  observation,  all  the  other 
elements  being  ?:nown,  we  were  to  calculate  either  what  mean  longitude 
the  planet  had  at  a  given  epochal  time,  or,  which  would  come  to  the 
same  thing,  at  what  moment  of  time  (thenceforward  to  be  assumed  as 
an  epoch)  it  had  a  given  mean  longitude.  It  is  evident  that  by  this 
means  the  epoch,  if  not  otherwise  known,  would  become  known,  whether 
we  consider  it  as  the  moment  of  time  corresponding  to  a  convenient  mean 
longitude,  or  as  the  mean  longitude  corresponding  to  a  convenient  time. 
The  latter  way  of  considering  it  has  some  advantages  in  respect  of  general 
convenience,  and  astronomers  are  in  agreement  in  employing,  as  an  ele- 


VARIATION   OP  THE   EPOCH. 


40" 


ment  under  the  title  "  Epoch  of  the  mean  longitude,"  the  meab  longitude 
of  the  planet  so  computed  for  a  fixed  date ;  aa,  for  instaL  the  commence 
went  of  the  year  1800,  mean  time  at  a  given  place.  Supposing  now  all 
elements  of  the  orbit  invariable,  if  vre  were  to  go  through  this  reverse 
process,  and  thus  ascertain  the  epoch  (so  defined)  from  any  number  of 
different  perfectly  correct  heliocentric  longitudes,  it  is  clear  we  should 
always  come  to  the  same  result.  One  and  the  same  "epoch"  would  come 
out  from  all  the  calculations. 

(730.)  Considering  then  the  "epoch"  in  this  light,  as  merely  a  result 
of  this  reversed  process  of  calculation,  and  not  as  the  direct  result  of  au 
observation  instituted  for  the  purpose  at  the  precise  epochal  moment  of 
time,  (which  would  be,  generally  speaking,  impracticable,)  it  might  be 
conceived  subject  to  variation  in  two  distinct  ways,  viz.  dependently  and 
independently.  Dependently  it  must  vary,  as  a  C'^cessary  consequence  of 
the  variation  of  the  other  elements ;  because,  if  setting  out  from  one  and 
the  same  observed  heliocentric  longitude  of  the  planet,  we  calculate  back 
to  t.  opoch  with  two  different  sets  of  intermediate  elements,  the  one  set 
consioting  of  those  which  it  had  immediately  before  its  arrival  at  that 
longitude,  the  other  that  which  it  takes  up  immediately  after  (i.  e.  with 
an  unvaried  and  varied  system),  we  cannot  (unless  by  singular  accident  of 
mutual  counteraction)  arrive  at  the  same  result ;  and  the  difference  of  the 
results  is  evidently  the  variation  of  the  epoch.  On  the  other  hand,  how- 
ever,  it  cannot  vary  independently ;  for  since  this  is  the  only  mode  in 
which  the  unvaried  and  varied  epochs  can  become  known,  and  as  both 
result  from  direct  processes  of  calculation  involving  only  given  data,  the 
results  can  only  differ  by  reason  of  the  difference  of  those  data.  Or  we 
may  argue  thus.  The  change  in  the  path  of  the  planet,  and  .its  place  in 
that  path  so  changed,  at  any  future  time  (supposing  it  to  undergo  no  fur* 
ther  variation),  are  entirely  owing  to  the  change  in  its  velocity  and  direc- 
tion, produced  by  the  disturbing  forces  at  the  point  of  disturbance ;  now 
these  latter  changes  (as  we  have  above  seen)  are  completely  represented  by 
the  momentary  change  in  the  situation  of  the  upper  focus,  taken  in  com- 
bination with  the  momentary  variation  in  the  plane  of  the  orbit;  and  these 
therefore  express  the  total  effect  of  the  disturbing  forces.  There  is,  there- 
fore, no  direct  and  specific  action  on  the  epoch  as  an  independent  variable. 
It  is  simply  left  to  accommodate  itself  to  the  altered  state  of  things  in  the 
mode  already  indicated. 

(731.)  Nevertheless,  should  the  effects  of  perturbation  by  inducing 
changes  on  these  other  elements  affect  the  mean  longitude  of  the  planet 
in  any  other  way  than  can  be  considered  as  properly  taken  account  of,  by 
the  varied  periodic  time  due  to  a  change  of  axis,  such  effects  must  be  re- 


i| 


).: 


:^:i 


,;5: 


;'r 


408 


OUTLINES  OF  ASTRONOMY. 


garded  as  inoidont  on  the  epoch.  This  is  the  oas"  with  a  very  curious 
class  of  perturbations  which  we  are  now  to  consider,  and  which  have  their 
origin  in  an  alteration  of  the  average  distance  at  which  the  disturbed  body 
is  found  at  every  instant  of  a  complete  revolution,  distinct  from,  and  not 
brought  about  by  the  variation  of  the  major  semi-axis,  or  momentary 
"  mean  distance"  which  is  an  imaginary  magnitude,  to  bo  carefully  diutiu- 
guiahod  from  the  avcrge  of  the  actual  distances  now  contemplated.  Per- 
turbations of  this  class  (like  the  moon's  variation,  with  which  they  arc 
intimately  connected)  are  independent  on  the  ezcentricity  of  the  disturbed 
orbit ;  for  which  reason  we  shall  simplify  our  treatment  of  this  part  of  the 
subject,  by  supposing  that  orbit  to  have  no  permanent  excentricity,  the 
upper  focus  in  its  successive  displacements  merely  revolving  about  a  mean 
position  coincident  with  the  lower.  We  shall  also  suppose  M  very  dis- 
tant, as  in  the  lunar  theory. 

(732.)  Referring  to  what  is  said  in  ar«s.  706  ai.'l  707,  and  to  the  figures 
accompanying  those  articles,  and  considering  first  the  effect  of  the  tangential 
force,  we  see  that  besides  the  effect  of  that  forn^  in  changing  the  length 
of  the  axis,  and  consequently  the  periodic  time,  it  causes  the  upper  focus 
H  to  describe,  in  each  revolution  of  P,  a  four-cusped  curve,  a,  6,  d,  e, 
about  S,  all  whose  intercuspidal  arcs  are  similar  and  equal.  This  supposes 
M  fixed,  and  at  an  invariable  distance, — suppositions  which  simplify,'  tbe 
relations  of  the  subject,  and  (as  wo  ;.^hali  afterwards  show)  do  not  affect 
the  general  nature  of  the  copclusiead  to  be  drawn.  In  virtue,  then,  of 
the  excentricity  thus  given  rise  to,  P  will  be  at  the  perigee  of  its  momen- 
tary ellipse  at  syzygies  and  in  its  apogee  at  quadratures.  Apart,  there/ore, 
from  tlie  change  arising  from  the  variation  of  axis,  the  distance  of  P 
from  S  will  be  less  at  syzygies,  and  greater  at  quadratures,  than  in  the 
original  circle.  But  the  average  of  all  the  distances  during  a  whole  revo- 
lution will  be  unaltered ;  because  the  distances  of  a,  d,  b,  e,  from  S  being 
equal,  and  the  arcs  symmetrical,  the  approach  in  and  about  perigee  will 
be  equal  to  the  recess  in  and  about  tbe  apogee.  And,  in  like  manner,  the 
effect  of  the  changes  going  on  in  the  length  of  the  axis  itself,  on  the 
average  in  question,  is  nil,  because  the  alternate  increases  and  decreases 
of  that  length  balance  each  other  iu  a  complete  revolution.  Thus  we  see 
that  the  tangential  force  is  exclud  4  from  all  influence  in  producing  the 
class  of  perturbations  now  under  consideration. 

(733.)  It  is  otherwise  with  respect  to  the  normal  force.  In  virtue  of 
the  action  of  that  force  the  upper  focus  describes,  in  each  revolution  of  P, 
the  four-cusped  curve  (fg.  art.  707),  whose  intercuspidal  arcs  are  alter- 
nately of  very  unequal  extent,  arising,  as  we  have  seen,  from  the  longer 
duration  and  greater  energy  of  the  outward  than  of  th6  inward  action  of 


INEQUALITIES  INCIDENT  ON  THE   hiOCH. 


409 


.  H 


a  we  see 


the  disturbing  force.  Although,  therefore,  in  perigee  at  eyzygios  and  in 
apogoc  nt quadratures,  the apogeal recess  is  much  greater  than  the  pcriget' 
approach,  inasmuch  as  S  d  greatly  exceeds  S  a.  On  the  average  of  u 
whole  revolution,  then,  the  recesses  will  preponderate,  ond  the  average 
distance  will  therefore  be  greater  in  the  disturbed  than  in  the  undisturbed 
orbit.  And  it  is  manifest  that  this  conclusion  is  quite  independent  of  any 
change  in  the  length  of  the  axis,  which  the  normal  force  has  no  power  to 
produce. 

(734.)  But  neither  does  the  normal  force  operate  ony  change  of  linear 
velocity  in  the  disturbed  body.  When  carried  out,  therefore,  by  the 
efTuct  of  that  force  to  a  greater  distance  from  S,  the  angular  velocity  of  its 
motion  round  S  will  be  diminished :  and  contrariwise  when  brought  nearer. 
The  avornge  of  all  the  momentary  angular  motions,  therefore,  will  de- 
creafic  with  the  increase  in  that  of  the  momentary  distances;  and  in  a 
higher  ratio,  since  the  angular  velocity,  under  an  equable  description  of 
areas,  is  inversely  as  the  square  of  the  distance,  and  the  disturbing  force, 
being  (in  the  case  supposed)  directed  to  or  from  the  centre,  docs  not  dis- 
turb that  equable  description  (art.  G16).  Consequently,  on  the  average 
of  a  whole  revolution,  the  angular  motion  is  slower,  and  therefore  the  time 
of  completing  a  revolution,  and  returning  to  the  same  longitude,  longer 
than  in  the  undisturbed  orbit,  and  tJiat  independent  of  and  without  any 
reference  to  the  length  of  the  momentary  axis,  and  the  "  periodic  time  " 
or  "mean  motion"  dependent  thereon.  We  leave  to  the  reader  to  follow 
out  (as  is  easy  to  do)  the  same  train  of  reasoning  in  the  cases  of  planetary 
perturbation,  when  M  is  not  very  remote,  and  when  it  is  interior  to  the 
disturbed  orbit.  In  the  latter  case  the  preponderant  effect  changes  from 
a  retardation  of  angular  velocity  to  an  acceleration,  and  tho  dilatation  of 
the  average  dimensions  of  F's  orbit  to  a  contraction. 

(735.)  The  above  is  an  accurate  analysis,  according  to  strict  dynamical 
principles,  of  an  effect  which,  speaking  roughly,  may  be  assimilated  to  an 
alteration  of  M's  gravitation  towards  S  by  the  mean  preponderant  amount 
of  the  outward  and  inward  action  of  the  normal  forces  constantly  exerted 
—  nearly  as  would  be  the  case  if  the  mass  of  the  disturbing  body  were 
formed  into  a  ring  of  uniform  thickness,  concentric  with  S,  and  of  such 
diameter  as  to  exert  an  action  on  P  everywhere  equal  to  such  mean  pre- 
ponderant force,  and  in  the  same  direction  as  to  inwards  or  outwards.  For 
it  is  clear  that  the  action  of  such  a  ring  on  P,  will  be  the  difference  of  its 
attractions  on  the  two  points  P  and  S,  of  which  the  latter  occupies  its 
centre,  the  former  is  excentric.  Now  the  attraction  of  a  ring  on  its 
centre  is  manifestly  equal  in  all  directions,  and  therefore,  estimated  in 
any  one  direction,  is  zero.     On  the  other  hand,  on  a  point  P  out  of  its 


i 


m 


410 


OUTLINES   OF  ASTRONOMY. 


>^ 


centre,  if  xciilUn  the  ring,  the  roaultiog  attraotioD  will  always  bo  outtmrdt, 
towurda  the  uearoat  point  of  the  ring,  or  directly  from  the  centre.'  But 
if  P  lie  without  the  ring,  the  resulting  force  will  act  always  mwanh, 
urging  P  towards  its  centre.  Ileuce  it  appears  that  the  mean  efTuut  of 
the  rudiul  forco  of  the  ring  will  be  different  in  its  direction,  according  as 
the  orbit  of  the  disturbing  body  is  exterior  or  interior  to  that  of  the  dis- 
turbed. In  the  former  case  it  will  act  in  diminution,  in  the  latter  in 
augmentation  of  the  central  gravity. 

(786.)  Regarding,  still,  only  the  mean  effect,  as  produced  in  a  great 
number  of  revolutions  of  both  bodies,  it  is  evident  that  such  an  increut^e 
of  central  force  will  be  accompanied  with  a  diminution  of  periodic  time 
and  distance  of  a  body  revolving  with  a  stated  velocity,  and  vice  vend. 
This,  as  We  have  shown,  is  the  first  and  most  obvioua  effect  of  the  rudial 
part  of  the  disturbing  force,  when  exactly  analyzed.  It  alters  permanently, 
and  by  a  certain  mean  amount,  the  distances  and  times  of  revolution  of 
all  the  bodies  composing  the  planetary  system,  from  what  they  would  be, 
did  each  planet  circulate  about  the  sun  uninfluenced  by  the  attraction  of 
the  rest;  the  angular  motion  of  the  interior  bodies  of  the  system  being 
thus  rendered  less,  and  those  of  the  exterior  greater,  than  on  that  suppo- 
sition. The  latter  effect,  indeed,  might  bo  at  once  concluded  from  this 
obvious  consideration,  —  that  all  the  planets  revolving  interiorly  to  any 


'  As  this  is  a  proposition  which  the  equilibrium  of  Saturn's  ring  renders  not  merely 
speculative  or  illustrative,  it  will  be  well  to  demonstrate  it ;  which  may  be  done  very 
simply,  and  without  the  aid  of  any  calculus.  Conceive  a  spherical  shell,  and  a  point 
within  it :  every  line  passing  through  the  point,  and  terminating  both  ways  in  the  shell, 
will,  of  course,  be  equally  incUned  to' its  surface  at  either  end,  being  a  chord  of  a  sphe- 
rical surface,  and  therefore  symmetrically  related  to  all  its  parts.  Now,  conceive  a 
small  double  cone,  or  pyramid,  having  its  apex  at  the  point,  and  formed  by  the  conicnl 
motion  of  such  a  line  round  the  point.  Then  will  the  two  portions  of  the  spherical 
•hell,  which  form  the  bases  of  both  the  cones,  or  pyramids,  be  similar  and  equally  in- 
clined to  their  axes.  Therefore  their  areas  will  be  to  each  other  as  the  squares  of  their 
distances  from  the  common  apex.  Therefore  their  attractions  on  it  will  be  equal,  be- 
cause the  attraction  is  as  the  attracting  matter  directly,  and  the  square  of  its  distance 
inversely.  Now,  these  attractions  act  in  opposite  directions,  and  therefore  coutitsract 
each  other.  Therefore  the  point  is  in  equilibrium  between  them  ;  and  as  the  siime  is 
true  of  every  such  pair  of  areas  into  which  the  spherical  shell  can  be  broken  up,  there- 
fore the  point  will  be  in  equilibrium  however  situated  within  such  a  spherical  iiheli. 
Now  take  a  ring,  and  treat  it  similarly,  breaking  its  circumference  up  into  pairs  of  ele- 
ments, the  baees  of  triangles  formed  by  lines  passing  through  the  attracted  point, 
Here  the  attracting  elements  being  2tne(,  not  surfaeet,  are  in  the  stm^j^e  ratio  of  the 
distances,  not  the  duplicate,  as  they  should  be  to  maintain  the  equilibrium.  Tlurefore 
it  will  not  be  maintained,  but  the  nearest  elements  will  have  the  superiority,  and  the 
point  will,  on  the  whole,  be  urged  towards  the  nearest  part  of  the  ring.  I'he  same  is 
true  of  every  linear  ring,  and  is  therefore  true  of  any  assemblage  of  concentric  ones 
forming  a  tlut  utiiiulos,  like  the  ring  of  Saturn. 


INEQUALITIES   INCIDENT   ON   THE   EPOCH. 


411 


orbit  mAv  be  cr^tdered  M  adding  to  the  general  n^t^ngute  of  (bs  attract' 
log  uiult4)r  wilblD,  whutb  i«  not  the  less  officiuiit  for  being  didtribuled  <rV9X 
ipace,  and  maiutained  iu  a  Htute  of  uirculation.  * 

(737.)  This  effect,  however,  is  one  which  we  have  no  means  of  metk- 
suriug,  or  oven  of  detecting,  otherwise  than  by  calculation.  For  our 
knowK  dge  of  the  periods  of  the  planets  is  drawn  from  observations  made 
on  them  in  their  actual  state,  and  therefore  under  the  influence  of  this, 
which  may  be  regarded  as  a  sort  of  constant  part  of  the  perturbative 
action.  Their  observed  mean  motionb  are  therefore  affected  by  the  whole 
amount  of  its  influence ;  and  we  have  no  means  of  distinguishing  this  by 
observation  from  the  direct  effect  of  tho  sun's  attraction,  with  which  it  is 
blended.  Our  knowledge,  however,  of  the  masses  of  the  planets  assures 
us  that  it  is  extremely  small ;  and  this,  in  fact,  is  all  which  it  is  at  all 
importauc  to  us  to  know,  in  the  theory  of  their  motions. 

(738.)  The  action  of  the  sun  upon  the  moon,  in  like  manner,  tends,  by 
its  mean  influence  during  many  successive  revolutions  of  both  bodies,  to 
increase  permanently  the  moon's  distance  and  periodic  time.  T^ut  this 
general  average  is  not  established,  either  in  the  case  of  tho  moon  or 
planets,  without  a  series  of  subordinate  fluctuations,  which  we  have  pur- 
posely neglected  to  take  account  of  in  the  above  reasoning,  and  which 
obviously  tend,  in  the  average  of  a  great  multitude  of  revolutions,  to 
neutralize  each  other.  In  the  lunar  theory,  however,  some  of  these  sub- 
ordinate fluctuations  are  very  sensible  to  observation.  The  most  conspi- 
cuous of  these  is  the  moon's  annual  equation ;  so  called  because  it  consists 
in  an  alternate  increase  and  decrease  in  her  longitude,  corresponding  with 


Fig.  106. 


■>n 


the  earth's  situation  in  its  annual  orbit ;  t.  e.  to  its  angular  distance  from 
the  perihelion,  and  therefore  having  a  year  instead  of  a  month,  or  aliquot 
part  of  a  mouth,  for  its  period.  To  understand  the  mode  of  its  produc- 
tion, let  us  suppose  the  sun,  still  holding  a  fixed  position  in  longitude,  to 
approach  gradually  nearer  to  the  earth.  Then  will  all  its  disturbing  forces 
be  gradually  increased  in  a  very  high  ratio  compared  with  the  diminution 
of  the  distance  (being  inversely  as  its  cube ;  so  that  its  effects  of  every 
kind  are  three  times  greater  in  respect  of  any  change  of  distance,  than 


412 


OUTLINES   OF   ASTRONOMY. 


they  would  be  by  the  simple  law  of  proportionality).  Hence,  it  is  obvious 
that  the  focus  H  (art.  707)  in  the  act  of  describing  each  intercuspidal  arc 
of  the  curve  a,  d,  h,  e,  will  be  continually  carried  out  farther  and  farther 
from  S ;  and  the  curve,  instead  of  returning  into  itself  at  the  end  of  each 
revolution,  will  open  out  into  a  sort  of  cuspidated  spiral,  as  in  the  figure 
annexed.  Retracing  now  the  reasoning  of  art.  733,  as  adapted  to  this 
state  of  things,  it  will  be  seen  that  so  long  as  this  dilatation  goes  on,  so 
long  will  the  difference  between  M's  recess  from  S  in  aphelio  and  its 
approach  in  perihelio  (which  is  equal  to  the  difference  of  consecutive  long 
and  short  semidiameters  of  this  curve)  also  continue  to  increase,  and  with 
it  the  average  of  the  distances  of  M  from  S  in  a  whole  revolution,  and 
consequently  also  the  time  of  performing  such  a  revolution.  The  reverse 
process  will  go  on  as  the  sun  again  recedes.  Thus  it  appears  that,  as  the 
sun  approaches  the  earth,  the  mean  angular  motion  of  the  moon  on  the 
average  of  a  whole  revolution  will  diminish,  and  the  duration  of  each 
lunation  will  therefore  exceed  that  of  the  foregoing,  and  vice  versd. 

(739.)  The  moon's  orbit  being  supposed  circular,  the  sun's  orbitual 
motion  will  have  no  other  effect  than  to  keep  the  moon  longer  under  the 
influence  of  every  gradation  of  the  disturbing  force,  than  would  have  been 
the  case  had  his  situation  in  longitude  remained  unaltered  (art,  711.)  The 
effects,  therefore,  will  take  place  only  on  an  increased  scale  in  the  propor- 
tion of  the  increased  time ;  i.  e.  in  the  proportion  of  the  synodic  to  the 
sidereal  revolution  of  the  moon.  Observation  confirms  these  results,  and 
assigns  to  the  inequality  in  question  a  maximum  value  of  between  10'  and 
11',  by  whieb.  the  moon  is  at  one  time  in  advance  of,  and  at  another  be- 
hind, its  mean  place,  in  consequence  of  this  perturbation. 

(740.)  To  this  class  of  inequalities  we  must  refer  one  of  great  import- 
ance, and  extending  over  an  immense  period  of  time,  known  by  the  _arae 
of  the  secular  acceleration  of  the  moon's  mean  motion.  It  had  been 
observed  by  Dr.  Halley,  on  comparing  together  the  records  of  the  most 
ancient  lunar  eclipses  of  the  Chaldean  astronomers  with  those  of  modern 
times,  that  the  period  of  the  moon's  revolution  at  present  is  sensibly 
shorter  than  at  that  remote  epoch ;  and  this  result  was  confirmed  by  a 
further  comparison  of  both  sets  of  observations  with  those  of  the  Arabian 
astronomers  of  the  eighth  and  ninth  centuries.  It  appeared,  from  these 
comparisons,  that  the  rate  at  which  the  moon's  mean  motion  increases  is 
about  11  seconds  per  century,  —  a  quantity  small  in  itself,  but  becoming 
considerable  by  its  accumulation  during  a  succession  of  ages.  This  re- 
markable fact,  like  the  great  equation  of  Jupiter  and  Saturn,  had  been 
long  the  subject  of  toilsome  investigation  to  geometers.  Indeed,  so 
difficult  did  it  appear  to  render  any  exact  account  of,  that  while  somi 


ANNUAL  INEQUALITY   OF  THE  MOON. 


418 


were  on  the  point  of  again  declaring  the  theory  of  gravity  inadequate  to 
its  explanation,  others  were  for  rejecting  altogether  the  evidence  on  which 
it  rested,  although  quite  as  satisfactory  as  that  on  which  most  historical 
events  are  credited.  It  was  in  this  dilemma  that  Laplace  once  more 
stepped  in  to  rescue  physical  astronomy  from  its  reproach,  by  pointing  out 
the  real  cause  of  the  phaenomenon  in  question,  which,  when  so  explained, 
is  one  of  the  most  curious  and  instructive  in  the  whole  range  of  our  sub- 
ject, —  one  which  leads  our  speculations  farther  into  the  past  and  future, 
and  points  to  longer  vistas  in  the  dim  perspective  of  changes  which  our 
system  has  undergone  and  is  yet  to  undergo,  than  any  other  which  obser- 
vation assisted  by  theory  has  developed. 

(741.)  The  year  is  not  an  exact  number  of  lunations.  It  consists  of 
twelve  and  a  fraction.  Supposing  then  the  sun  and  moon  to  set  out  from 
conjunction  together ;  at  the  twelfth  conjunction  subsequent  the  sun  will 
not  have  returned  precisely  to  the  same  point  of  its  annual  orbit,  but  will 
fall  somewhat  short  of  it,  and  at  the  thirteenth  will  have  overpassed  it. 
Hence  in  twelve  lunations  the  gain  of  longitude  during  the  first  half  year 
will  be  somewhat  under  and  in  thirteen  somewhat  over-compensated.  In 
twenty-six  it  will  be  nearly  twice  as  much  over-compensated,  in  thirty-nine 
not  quite  so  nearly  three  times  as  much,  and  so  on,  until,  after  a  certain 
number  of  such  multiples  of  a  lunation  have  elapsed,  the  sun  will  be 
found  half  a  revolution  in  advance,  and  in  place  of  receding  farther  at  the 
expiration  of  the  next,  it  will  have  begun  to  approach.  From  this  time 
every  succeeding  cycle  will  destroy  some  portion  of  that  over-compensa- 
tion, until  a  complete  revolution  of  the  sun  in  excess  shall  be  accom- 
plished. Thus  arises  a  subordinate  or  rather  supplementary  inequality, 
having  for  its  period  as  many  years  as  is  necessary  to  multiply  the  defi- 
cient arc  into  a  whole  revolution,  at  the  end  of  which  time  a  much  more 
exact  compensation  will  have  been  operated,  and  so  on.  Thus  after  a 
moderate  number  of  years  an  almost  perfect  compensation  will  be  effected, 
and  if  we  extend  our  views  to  centuries  we  may  consider  it  as  quite  so. 
Such  at  least  would  be  the  case  if  the  solar  ellipse  were  invariable.  But 
that  ellipse  is  kept  in  a  continual  but  excessively  slow  state  of  change  by 
the  action  of  the  planets  on  the  earth.  Its  axis,  it  is  true,  remains  unal- 
tered; but  its  excentricity  is,  and  has  been  since  the  earliest  ages,  dimin- 
ishing ;  and  this  diminution  will  continue  (there  is  little  reason  to  doubt) 
till  the  excentricity  is  annihilated  altogether,  and  the  earth's  orbit  becomes 
a  perfect  circle ;  after  which  it  will  again  open  out  into  an  ellipse,  the 
excentricity  will  again  increase,  attain  a  certain  moderate  amount,  and 
then  again  decrease.  The  time  required  for  these  evolutions,  though 
calculable,  has  not  been  calculated,  further  than  to  satisfy  us  that  it  itt 


r, 


'\ 


i ' 


I,  I 
1  ■ 


lil 


A  1 


»U  -If 


■fillil 


15, 


414 


OUTLINES   OF  ASTRONOMY. 


not  to  be  reckoned  by  hundreds  or  by  thousands  of  years.  It  is  a  period, 
in  short,  in  which  the  whole  history  of  astronomy  and  of  the  human  race 
occupies  but  as  it  were  a  point,  during  which  all  its  changes  are  to  be 
regarded  as  uniform.  Now,  it  is  by  this  variation  in  the  excentricity  of 
the  earth's  orbit  that  the  secular  acceleration  of  the  moon  is  caused.  The 
compensation  above  spoken  of  (even  after  the  lapse  of  centuries)  will  now, 
we  see,  be  only  imperfectly  effected,  owing  to  this  slow  shifting  of  one 
of  the  essential  data.  The  steps  of  restoration  are  no  longer  identical 
with,  nor  equal  to,  those  of  change.  The  struggle  up  hill  is  not  main- 
tained on  equal  terms  with  the  downward  tendency.  The  ground  is  all 
the  while  slowly  sliding  beneath  the  feet  of  the  antagonists.  During  the 
whole  time  that  the  earth's  excentricity  is  diminishing,  a  preponderance 
is  given  to  the  reaction  over  the  action ;  and  it  is  not  till  that  diminution 
shall  cease,  that  the  tables  will  be  turned,  and  the  process  of  ultimate 
restoration  will  commence.  Meanwhile,  a  minute,  outstanding,  and  un- 
compensated effect  in  favour  of  acceleration  is  left  at  each  recurrence,  or 
near  recurrence,  of  the  same  configurations  of  the  sun,  the  moon,  and  the 
solar  perigee.  These  accumulate,  and  at  length  affect  the  moon's  longi- 
tude to  an  extent  not  to  be  overlooked. 

(742.)  The  phaanomenon,  of  which  we  have  now  given  an  account,  is 
another  and  very  striking  example  of  the  propagation  of  a  periodic  change 
from  one  part  of  a  system  to  another.  The  planets,  with  one  exception, 
have  no  direct  appreciable  action  on  the  lunar  motions  as  referred  to  the 
earth.  Their  masses  are  too  small,  and  their  distances  too  great,  for  their 
difference  of  action  on  the  moon  and  earth  ever  to  become  sensible.  Yet 
their  effect  on  the  earth's  orbit  is  thus,  we  see,  propagated  through  the 
sun  to  that  of  the  moon ;  and,  what  is  very  remarkable,  the  transmitted 
effect  thus  indirectly  produced  on  the  angle  described  by  the  moon  round 
the  earth  is  more  sensible  to  observation  than  that  directly  produced  by 
them  on  the  angle  described  by  the  earth  round  the  sun. 

(743.)  Referring  to  the  renaoning  of  art.  738,  we  shall  perceive  that 
if,  owing  to  any  other  cause  than  its  elliptic  motion,  the  sun's  distance 
from  the  earth  be  subject  to  a  periodical  increase  and  decrease,  that  varia- 
tion will  give  rise  to  a  lunar  inequality  of  equal  period  analogous  to  the 
annual  equation.  It  thus  happens  that  very  minute  changes  impressed 
on  the  orbit  of  the  earth,  by  the  direct  action  of  the  planets,  (provided 
their  periods,  though  not  properly  speaking  secular,  be  of  considerable 
length,)  may  make  themselves  sensible  in  the  lunar  motions.  The  longi- 
tude of  that  satellite,  as  observed  from  the  earth,  is,  in  fact,  singularly 
sensible  to  this  kind  of  reflected  action,  which  illustrates  in  a  striking 
manner  the  principle  of  forced  vibrations  laid  down  in  art.  650.    Th» 


INDIRECT  ACTION   OF   VENUS   ON  THE   MOON. 


415 


reason  of  this  will  be  readily  apprehended,  if  we  consider  that  however 
trifling  the  increase  of  her  longitude  which  would  arise  in  a  single  revolu- 
tion, from  a  minute  and  almost  infinitesimal  increase  of  her  mean  angu- 
lar velocity,  that  increase  is  not  only  repeated  in  each  subsequent  revolu- 
tion, but  is  reinforced  during  each  by  a  similar  fresh  accession  of  angular 
motion  generated  in  its  lapse.  This  process  goes  on  so  long  as  the  angu- 
lar motion  continues  to  increase,  and  only  begins  to  be  reversed  when 
lapse  of  time,  bringing  round  a  contrary  action  on  the  angular  motion, 
shall  have  destroyed  the  excess  of  velocity  previously  gained,  and  begun 
to  operate  a  retardation.  In  this  respect,  the  advance  gained  by  the  moon 
on  her  undisturbed  place  may  be  assimilated,  during  its  increase,  to  the 
space  described  from  rest  under  the  action  of  a  continually  accelerating 
force.  The  velocity  gained  in  each  instimt  is  not  only  effective  in  carry- 
ing the  body  forward  during  each  subsequent  instant,  but  new  velocities 
are  every  instant  generated,  and  go  on  adding  their  cumulative  effects  to 
those  before  produced. 

(744.)  The  distance  of  the  earth  from  the  sun,  like  that  of  the  moon 
from  the  earth,  may  be  affected  in  its  average  value  estimated  over  long 
periods  embracing  many  revolutions,  in  two  modes,  conformably  to  the 
theory  above  delivered.  1st,  it  may  vary  by  a  variation  in  the  length  of 
the  axis  major  of  its  orbit,  arising  from  the  direct  action  of  some  tangen- 
tial disturbing  force  on  its  velocity,  and  thereby  producing  a  change  of 
mean  motion  and  periodic  time  in  virtue  of  the  Keplerian  law  of  periods, 
which  declares  that  the  periodic  times  are  in  the  sesquiplicate  ratio  of  the 
mean  distances.  Or,  2dly,  it  may  vary  by  reason  of  that  peculiar  action 
on  the  average  of  actual  distances  during  a  revolution,  which  arises  from 
variations  of  excentricity  and  perihelion  only,  and  which  produces  that 
sort  of  change  in  the  mean  motion  which  we  have  characterized  as  inci- 
dent on  the  epoch.  The  change  of  mean  motion  thus  arising,  has  nothing 
whatever  to  do  with  any  variation  of  the  major  axis.  It  does  not  depend 
on  the  change  of  distance  by  the  Keplerian  law  of  periods,  but  by  that 
of  areas.  The  altered  mean  motion  is  not  sub-sesquiplicate  to  the  altered 
axis  of  the  ellipse,  which  in  fact  does  not  alter  at  all,  but  is  sub-dupli- 
cate to  the  altered  average  of  distances  in  a  revolution ;  a  distinction 
which  must  be  carefully  borne  in  mind  by  every  one  who  will  clearly  un- 
derstand either  the  subject  itself,  or  the  force  of  Newton's  explanation 
of  it  in  the  6th  Corollary  of  his  celebrated  66th  Proposition.  In  which- 
ever mode,  however,  an  alteration  in  the  mean  motion  is  effected,  if  we 
accommodate  the  general  sense  of  our  language  to  the  specialities  of  the 
case,  it  remains  true  that  every  change  in  the  mean  motion  is  accompa- 
nied with  a  corresponding  change  in  the  mean  distance. 


t"  [• 

1 

\ 

,1 

f  ' ! 

\ 

'   M. 


416 


OUTLINES  OF  ASTRONOMY. 


(745.)  Now  we  have  seen  (art.  726),  that  Venus  produces  in  the  earth 
a  perturbation  in  lengitude,  of  so  long  a  period  (240  years),  that  it  can- 
not well  be  regarded  without  violence  to  ordinary  language,  otherwise  than 
as  an  equation  of  the  mean  motion.  Of  course,  therefore,  it  follows  that 
during  that  half  of  this  long  period  of  time,  in  which  the  earth's  motion 
is  retarded,  the  distance  between  the  sun  and  earth  is  on  the  increase,  and 
vice  versd.  Minute  as  is  the  equation  in  question,  and  consequent  altera- 
tion of  solar  distance,  and  almost  inconceivably  minute  as  is  the  effect 
produced  on  the  moon's  mean  angular  velocity  in  a  single  lunation,  yet 
the  great  number  of  lunations  (1484),  during  which  the  effect  goes  on 
accumulating  in  one  direction,  causes  the  moon,  at  the  moment  when  that 
accumulation  has  attained  its  maximum  to  be  very  sensibly  in  advance  of 
its  undisturbed  place  (viz.  by  23"  of  longitude),  and  after  1484  more 
lunations,  as  much  in  arrear.  The  calculations  by  which  this  curious 
result  has  been  established,  formidable  from  their  length  and  intricacy, 
are  due  to  the  industry,  as  the  discovery  of  its  origin  is  to  the  sagacity, 
of  Professor  Hansen. 

(746.)  The  action  of  Venus,  just  explained,  is  indirect,  being  as  it 
were  a  sort  of  reflection  of  its  influence  on  the  earth's  orbit.  But  a  very 
remarkable  instance  of  its  influence,  in  actually  perturbing  the  moon's  mo- 
tions by  its  direct  attraction,  has  been  pointed  out,  and  the  inequality  due 
to  it  computed  by  the  same  eminent  geometer.'  As  the  details  of  his 
processes  have  not  yet  appeared,  we  can  here  only  explain,  in  general 
terms,  the  principle  on  which  the  result  in  question  depends,  and  tho 
nature  of  the  peculiar  adjustment  of  the  mean  angular  velocities  of  the 
earth  and  Venus  which  render  it  effective.  The  disturbing  forces  of 
Venus  on  the  moon  are  capable  of  being  represented  or  expressed  (as  is 
indeed  generally  the  case  with  all  the  forces  concerned  in  producing  pla- 
netary disturbance)  by  the  substitution  for  them  of  a  series  of  other  forces, 
each  having  a  period  or  cycle  within  which  it  attains  a  maximum  in  one 
direction,  decreases  to  nothing,  reverses  its  action,  attains  a  maximum  in 
the  opposite  direction,  again  decreases  to  nothing,  again  reverses  its  action, 
and  reattains  its  forraer  magnitude,  and  so  on.  These  cycles  differ  for 
each  particular  constituent  or  term,  as  it  is  called,  of  the  total  forces  con- 
sidered as  so  broken  up  into  partial  ones,  and,  generally  speaking,  every 
combination  which  can  be  formed  by  subtracting  a  multiple  of  the  mean 
motion  of  one  of  the  bodies  concerned  from  a  multiple  of  that  of  the 
other,  and  when  there  are  three  bodies  disturbing  one  another,  every  such 
triple  combination  becomes,  under  the  technical  name  of  an  argument, 
the  cyclical  representative  of  a  torce  acting  in  the  manner  and  according 
'  Astronomische  Nacbrichten,  No.  597. 


DIRECT  ACTION   OP   VENUS   ON  THE   MOON. 


417 


to  the  law  described.     Each  of  these  periodically  acting  forces  produces 
its  perturbative  eflPect,  according  to  the  law  of  the  superposition  of  small 
motions,  as  if  the  others  had  no  existence.     And  if  it  happen,  as  in  an 
immense  majority  of  cases  it  does,  that  the  cycle  of  any  particular  one  of 
these  partial  forces  has  no  relation  to  the  periodic  time  of  the  disturbed 
body,  so  as  to  bring  it  to  the  same,  or  very  nearly  the  same  point  of  its 
orbit,  or  to  any  situation  favourable  to  any  particular  form  of  disturbance, 
over  and  over  again  when  the  force  is  at  its  maximum ;  that  force  will,  in 
a  few  revolutions,  neutralise  its  own  effect,  and  nothing  but  fluctuations 
of  brief  duration  can  result  from  its  action.     The  contrary  will  evidently 
be  the  case,  if  the  cycle  of  the  force  coincide  so  nearly  with  the  cycle  of 
the  moon's  anomalistic  revolution,  as  to  bring  round  the  maximum  of  the 
force  acting  in  one  and  the  same  direction  (whether  tangential  or  normal) 
either  accurately,  or  very  nearly  indeed  to  some  definite  point,  as,  for  ex- 
ample, the  apogee  of  her  orbit.     Whatever  the  eiFect  produced  by  such  a 
force  on  the  angular  motion  of  the  moon,  if  it  be  not  exactly  compen- 
sated in  one  cycle  of  its  action,  it  will  go  on  accumulating,  being  repeated 
over  and  over  again  under  circumstances  very  nearly  tho  same,  for  many 
successive  revolutions,  until  at  length,  owing  to  the  want  of  precise  accu- 
racy in  the  adjustment  of  that  cycle  to  the  anomalistic  period,  the  maxi- 
mum of  the  force  (in  the  same  phase  of  its  action)  is  brought  to  coincide 
with  a  point  in  the  orbit  (as  the  perigee),  determinative  of  an  opposite 
effect,  and  thus,  at  lecgtb,  a  compensation  will  be  worked  out ;  in  a  time, 
however,  so  much  the  longer  as  the  difference  between  the  cycle  of  the 
force  and  the  moon's  anomalistic  period  is  less. 

(747.)  Now,  in  fact,  in  the  case  of  Venus  disturbing  the  moon,  there 
exista  a  cyclical  combination  of  this  kind.  Of  course  the  disturbing  force 
of  Venus  on  the  moon  varies  with  her  distance  fiom  the  earth,  and  this 
•listance  again  depends  on  her  configuration  with  respect  to  the  earth  and 
the  sun,  taking  into  account  the  ellipticity  of  both  their  orbits.  Among 
the  combinations  which  take  their  rise  from  this  latter  consideration,  and 
which,  as  may  easily  be  supposed,  are  of  great  complexity,  there  is  a  term 
(an  exceediiigly  minute  one),  whose  argument  or  cycle  is  determined  by 
subtracting  16  times  the  mean  motion  of  the  earth  from  18  times  that  of 
Venus.  The  difference  is  so  very  nearly  the  mean  motion  of  the  moon 
in  L.i-  anomalistic  revolution,  that  whereas  the  latter  revolution  is  com- 
pleted in  27*  13"  18-  82 -S*,  the  cycle  of  the  forc<^  is  completed  in  27* 
IS""  7™  35-6»,  differing  from  the  other  by  no  more  than  10"  56-7',  or 
about  one  3625th  part  of  a  complete  period  of  the  moon  from  apogee  to 
apogee.  During  half  of  this  very  long  interval  (that  is  to  say,  during 
about  136i  years),  the  perturbations  produced  by  a  force  of  this  character, 
27 


4   ' 


r    .: 


418 


OUTLINES   OF  ASTRONOMY. 


go  on  increasing  and  accumulating,  and  are  destroyed  in  another  equal  in. 
terval.  Although  therefore  excessively  minute  in  their  actual  effect  on 
the  angular  motion,  this  minuteness  is  compensated  by  the  number  of  re- 
peated acts  of  accnmulation,  and  by  the  length  of  time  during  which  they 
continue  to  act  on  the  longitude.  Accordiugly  M.  Hansen  has  found  the 
total  amount  of  fluctuation  to  and  fro,  or  the  value  of  the  equation  of  the 
moon's  longitude  so  arising,  to  be  27 •4".  It  is  exceedingly  interesting  to 
observe  that  the  two  equations  considered  in  these  latter  paragraphs, 
account  satisfactorily  for  the  only  remaining  material  differences  between 
tlieory  and  observation  in  the  modern  history  of  this  hitherto  rebellious 
siatellite.  We  have  not  thought  it  necessary  (indeed  it  would  have  required 
a  treatise  on  the  subject)  to  go  into  a  special  account  of  the  almost  innu- 
merable other  lunar  inequalities  which  have  been  computed  and  tabulated, 
and  which  are  necessary  to  De  taken  into  account  in  every  computation 
of  her  place  from  the  tables.  Many  of  them  are  of  very  much  larger 
amount  than  these.  We  ought  not,  however,  to  pass  unnoticed,  that  the 
parallactic  inequality,  already  explained  (art.  712),  is  interesting,  as  afford- 
ing a  measure  of  the  sun's  distance.  For  this  equation  originates,  as  there 
shown,  in  the  fact  that  the  disturbing  forces  are  not  precisely  alike  in  the 
two  halves  of  the  moon's  orbit  nearest  to  and  most  remote  from  the  sun, 
all  their  values  being  greater  in  the  former  half.  As  a  knowledge  of  the 
relative  dimensions  of  the  solar  and  lunar  orbiu  enables  us  to  calculate  d 
priori,  the  amount  of  this  inequality,  so  a  knowledge  of  that  amount 
deduced  by  the  comparison  of  a  great  number  of  observed  places  of  the 
moon  with  tables  in  which  every  inequality  but  this  should  be  included, 
would  enable  us  conversely  to  ascertain  the  ratio  of  the  distances  in  ques- 
tion. Owing  to  the  smallness  of  the  inequality,  this  is  not  a  very  accu- 
rate mode  of  obtaining  an  element  of  so  much  importance  in  astronomy 
as  the  sun's  distance,  but  were  it  larger  (i.  e.  were  the  moon's  orbit  con- 
siderably larger  than  it  actually  is),  this  would  be,  perhaps,  the  most 
exact  method  of  any  by  which  it  could  be  concluded. 

(748.)  The  greatest  of  all  the  lunar  inequalities,  produced  by  pertur- 
bation, is  that  called  the  evection.  It  arises  directly  from  the  variation 
of  the  excentricity  of  her  orbit,  and  from  the  fluctuation  to  and  fro  in  the 
general  progress  of  the  line  of  apsides,  caused  by  the  different  situation 
of  the  sun,  with  respect  to  that  line  (arts.  685,  691).  Owing  to  these 
causes  the  moon  is  alternately  in  advance,  and  in  arrear  of  her  elliptic 
place  by  about  1°  20'  20".  This  equation  was  known  to  the  an(;ients, 
having  been  discovered  by  Ptolemy,  by  the  comparison  of  a  long  series 
of  observations,  handed  down  to  him  from  the  earliest  ages  of  astronomy. 
The  mode  in  which  the  effects  of  these  several  sources  of  inequality  be- 


I 


EFFECT  OF  THE  EARTH'S   SPHEROIDAL  FIGURE. 


419 


come  grouped  together  under  one  principal  argument,  common  to  them 
all,  belongs,  for  its  expk  -vtion,  rather  to  works  sjtecially  treating  of  the 
lunar  theory  than  to  a  treatise  of  this  kind. 

(749.)  Some  small  perturbations  are  produced  in  the  lunar  orbit  by 
the  protuberant  matter  of  the  earth's  equator.  The  attraction  of  a  sphere 
is  the  same  as  if  all  its  matter  were  condensed  into  a  point  in  its  centre ; 
but  that  is  not  the  case  with  a  spheroid.  The  attraction  of  such  a  mass 
is  neither  exactly  directed  to  its  centre,  nor  does  it  exactly  follow  the  law 
of  the  inverse  squares  of  the  distances.  Hence  will  arise  a  series  of 
perturbations,  eytremely  small  in  amount,  but  still  perceptible  in  the 
lunar  motions,  by  which  the  node  and  the  apogee  will  be  aflFected.  A 
more  remarkable  consequence  of  this  cause,  however,  is  a  small  nutation 
of  the  lunar  orbit,  exactly  analogous  to  that  which  the  moon  causes  in 
the  plane  of  the  earth's  equator,  by  its  action  on  the  same  elliptic  protu- 
berance. And,  in  general,  it  may  be  observed,  that  in  the  systems  of 
planets  which  have  satellites,  the  elliptic  figure  of  the  primary  has  a  ten- 
dency to  bring  the  orbits  of  the  satellites  to  coincide  with  its  equator, — a 
tendency  which,  though  small  in  the  case  of  the  earth,  yet  in  that  of  Jupiter, 
whose  ellipticity  is  very  considerable,  and  of  Saturn  especially,  where  the 
ellipticity  of  the  body  is  reinforced  by  the  attraction  of  the  rings,  becomes 
predominant  over  every  external  and  internal  cause  of  disturbance,  and 
produces  and  maintains  an  almost  exact  coincidence  of  the  planes  in 
question.  Such,  at  least,  is  the  case  with  the  nearer  satellites.  The 
more  distant  are  comparatively  less  affected  by  this  cause,  the  difference 
of  attractions  between  a  sphere  and  spheroid  diminishing  with  great  ra- 
pidity as  \\^e  distance  increases.  Thus,  while  the  orbits  of  all  the  interior 
satellites  of  Saturn  lie  almost  exactly  in  the  plane  of  the  ring  and  equator 
of  the  planet,  that  of  the  external  satellite,  whose  distance  from  Saturn 
is  between  sixty  and  seventy  diameters  of  the  planet,  is  inclined  to  that 
plane  considerably.  On  the  other  hand,  this  considerable  distance,  while 
it  permits  the  satellite  to  retain  its  actual  inclination,  prevents  (by  parity 
of  reasoning)  the  ring  and  equator  of  the  planet  from  being  perceptibly 
disturbed  by  its  attraction,  or  being  siibjected  to  any  appreciable  move- 
ments analogous  to  our  nutation  and  precession.  If  such  exist,  they 
must  be  much  slower  than  those  of  the  earth ;  the  mass  of  this  satellite 
being,  as  far  as  can  be  judged  by  its  apparent  size,  a  much  smaller  frac- 
tion of  that  of  Sat'^rn  than  the  moon  is  of  the  earth ;  while  the  solar 
precession,  by  reason  of  the  immense  distance  of  the  sun,  must  be  quite 
imperceptible. 

(750.)  The  subject  of  the  tides,  though  rather  belonging  to  terrestrial 
physics  than  properly  to  astronomy,  is  yet  so  directly  connected  with  the 


]^  ':Vi 


'I 

!  'I, 


I 

m 

■IH'- 


s        >.  I 


''m 


'# 


420 


OUTLINES  OF  ASTRONOMY. 


theory  of  the  lunax  porturbatioQs,  that  we  cannot  omit  some  explanatory 
notice  of  it,  especially  since  many  persons  find  a  strange  difficulty  in  con- 
ceiving the  manner  in  which  they  are  produced.  That  the  sun,  or  moon, 
should  by  its  attraction  heap  up  the  waters  of  the  ocean  under  it,  seems 
to  them  very  natural.  That  it  should  at  the  same  time  heap  them  up  on 
the  opposite  side  seems,  on  the  contrary,  palpably  absurd.  The  error  of 
this  class  of  objectors  is  of  the  same  kind  with  that  noticed  in  art.  723, 
and  consists  in  disregarding  the  attraction  of  the  disturbing  body  on  the 
mass  of  the  earth,  and  looking  on  it  as  wholly  effectiye  on  the  superficial 
water.  Were  the  earth  indeed  absolutely  fixed,  held  in  its  place  by  an 
external  force,  and  the  water  left  free  to  move,  no  doubt  the  effect  of  the 
disturbing  power  would  be  to  produce  a  single  accumulation  vertically 
under  the  disturbing  body.  But  it  is  not  by  its  whole  attraction,  but  by 
the  difference  of  its  attractions  on  the  superficial  water  at  both  sides,  and 
on  the  central  mass,  ^liat  the  waters  are  raised  *  just  as  in  the  theory  of 
the  moon,  the  difference  of  the  sun's  attractions  on  the  moon  and  on  the 
earth  (regarded  as  moveable  and  as  obeying  that  amount  of  attraction 
which  b  due  to  its  situation)  gives  rise  to  a  relative  tendency  in  the  mooa 
to  recede  from  the  earth  in  conjunction  and  opposition,  and  to  approach 
it  in  quadratures.  Referring  to  the  figure  of  art.  675,  instead  of  sup- 
posing A  D  B  C  to  represent  tbo  moon's  orbit,  let  it  be  supposed  to  repre- 
sent a  section  of  the  (comparatively)  thin  film  of  water  reposing  on  the 
globe  of  the  earth,  in  a  great  circle,  the  plane  of  which  passes  through 
the  disturbing  body  M,  which  we  shall  suppose  to  be  the  moon.  The 
disturbing  force  on  a  particle  at  P  will  then  (exactly  as  in  the  lunar 
theory)  be  represented  in  amount  and  direction  by  N  S,  on  the  same  scale 
on  which  S  M  represents  the  moon's  whole  attraction  on  a  particle  situ- 
ated at  S.  This  force,  applied  at  P,  will  urge  it  in  the  direction  P  X 
parallel  to  N  S ;  and  therefore,  when  compounded  with  the  direct  force  of 

Fig.  1C8. 


gravity  whioh  (neglecting  as  of  no  account  in  this  theivy  the  spheroidal 


OF  THE  TIDES. 


421 


form  0.^  the  earth)  urges  P  towards  S,  will  he  equivalent  to  a  single  force 
deviating  from  the  direction  PS  towards  X.     Suppose  PT  to  be  the  di- 
rection of  this  forcn,  which,  it  is  easy  to  see,  will  be  directed  towards  a 
point  in  D  S  pro^^ .  -rf,  at  an  extremely  small  distance  below  S,  because 
of  the  excessive  minuteness  of  the  disturbing  force  compared  to  gravity.' 
Then  if  this  be  done  at  every  point  of  the  quadrant  A  D,  it  will  be  evi- 
dent that  the  direction  P  T  of  the  resultant  force  will  be  always  that  of  a 
tangent  to  the  small  cuspidated  curve  a  d  at  T,  to  which  tangent  the  sur- 
face of  the  ocean  at  P  must  everywhere  be  perpendicular,  by  reason  of 
that  law  of  hydrostatics  which  requires  the  direction  of  gravity  to  be 
everywhere  perpendicular  to  the  durface  of  a  fluid  in  equilihrio.     The 
form  of  the  curve  D  P  A,  to  which  the  surface  of  the  ocean  will  tend  to 
conform  itself,  so  as  to  place  itself  everywhere  in  equilibrio  under  two 
acting  forces,  will  be  that  which  always  has  P  T  for  its  radius  of  curva- 
ture.    It  will  therefore  be  slightly  less  curved  at  D,  and  more  so  at  A, 
being  in  fact  no  other  than  an  ellipse,  having  S  for  its  centre,  da  for  its 
evolute,  and  S  A,  S  D  for  its  longer  and  shorter  semi-axes  respectively ;  so 
that  the  whole  surface  (supposing  it  covered  with  water)  will  tend  to  as- 
sume, as  its  form  of  equilibrium,  that  of  an  oblongated  ellipsoid,  having 
its  longer  axis  directed  towards  the  disturbing  body,  and  its  shorter  of 
course  at  right  angles  to  that  direction.     The  difference  of  the  longer  and 
shorter  semi-axes  of  this  ellipsoid  due  to  the  moon's  attraction  would  be 
about  58  inches :  that  of  the  ellipsoid,  similarly  formed  in  virtue  of  the 
sun,  about  2  J  times  less,  or  about  23  inches. 

(751.)  Let  us  suppose  the  moon  only  to  act,  and  to  have  no  orbitual 
motion ;  then  if  the  earth  also  had  no  diurnal  motion,  the  ellipsoid  of 
equilibrium  would  be  quietly  formed,  and  all  would  be  thenceforward 
tranquil.  There  is  never  time,  however,  for  this  spheroid  to  be  fully 
formed.  Before  the  waters  can  take  their  level,  the  moon  has  advanced 
in  her  orbit,  both  diurnal  and  monthly,  (for  in  this  theory  it  will  answer 
the  purpose  of  clearness  better,  if  we  suppose  the  earth's  diurnal  motion 
transferred  io  the  sun  and  moon  in  the  contrary  direction,)  the  vertex  of 
the  spheroid  has  shifted  on  the  earth's  surface,  and  the  ocean  has  to  seek 
a  new  bearing.  The  effect  is  to  produce  an  immensely  broad  and  exces- 
sively flat  wave  (not  a  circulating  current),  which  follows,  or  endeavours 

'  According  to  Newton's  calculation,  the  maximum  diHturbing  force  of  the  sun  on  the 
v/ater  does  not  exceed  one  2373G400th  part  of  ily  graviiy.  That  of  the  moon  will 
therefore  be  to  this  fraction  as  the  cube  of  the  sun's  distance  to  that  of  the  moon's  di- 
rectly, and  as  the  mass  of  the  sun  to  that  of  the  moon  inversely,  t.  e.  as  (400)' X  0  012517 
:  354936,  which,  reduced  to  numbers,  gives,  for  the  moon's  maximum  of  power  to  dis- 
turb  the  waters,  about  one  11400000th  of  gravity,  or  somewhat  less  than  2^  times  tbo 
Bun's. 


«'  i 


ij  -Jl 
■  I 

ft)  I 


:ii ; 


(    M 


to 


'• 


i. 


(I 


422 


OUTLINES   OF   ASTRONOMY. 


to  follow,  the  apparent  niotions  of  the  moon,  and  must,  in  fact,  by  tbo 
principle  of  forced  vibrations,  iniituto,  by  eijuul  though  not  by  nyiichronom 
periods,  all  the  periodical  inetjuulitics  of  that  motion.  When  the  hij^her 
or  lower  parts  of  this  wave  strike  our  coasts,  they  experience  what  we 
call  high  and  low  water. 

(752.)  The  sun  also  produces  precisely  such  a  wave,  whose  vertex  tends 
to  follow  the  apparent  motion  of  the  sun  in  the  heavens,  and  also  to  imi- 
tate its  periodic  inequalities.  This  solar  wave  co-exists  with  the  lunar — 
is  sometimes  superposed  on  it,  sometimes  transverse  to  it,  so  as  to  partly 
neutralize  it,  according  to  the  monthly  syuodical  configuration  of  the  two 
luminaries.  This  alternate  mutual  reinforcement  and  destruction  of  the 
solar  and  lunar  tides  cause  what  are  called  the  spring  and  neap  tides — the 
former  being  their  sum,  the  latter  their  difference.  Although  the  real 
amount  of  either  tide  is,  at  present,  hardly  within  the  reach  of  exact  cal- 
culation, yet  their  proportion  at  any  one  place  is  probably  not  very  remote 
from  that  of  the  ellipticities  which  would  belong  to  their  respective  sphe- 
roids, oould  an  equilibrium  be  attained.  Now  these  ellipticities,  for  the 
solar  and  lunar  spheroids,  are  respectively  about  two  and  five  feet ;  so  that 
the  average  spring  tide  will  be  to  the  neap  as  7  to  3,  or  thereabouts. 

(703.)  Another  effect  of  the  combination  of  the  solar  and  lunar  tides 
is  what  is  called  the  primin<j  and  lagging  of  the  tides.  If  the  moon 
alone  existed,  and  moved  in  the  plane  of  the  equator,  the  tide-day  (<'.  e. 
the  interval  between  two  successive  arrivals  at  the  same  place  of  the  same 
vertex  of  the  tide-wave)  would  be  the  lunar  day  (art.  143),  formed  by 
the  combination  of  the  moon's  sidereal  period  and  that  of  the  earth's 
diurnal  motion.  Similarly,  did  the  sun  alone  exist,  and  move  always  on 
the  equator,  the  tide-day  would  be  the  mean  solar  day.  The  actual  tide- 
day,  then,  or  the  interval  of  the  occurrence  of  two  successive  maxima  of 
their  superposed  waves,  will  vary  as  the  separate  waves  approach  to  or  re- 
cede from  coincidence ;  because,  when  the  vertices  of  two  waves  do  not 
coincide,  their  joint  height  has  its  maximum  at  a  point  intermediate  be- 
tween them.  This  variation  from  uniformity  in  the  lengths  of  successive 
tide-days  is  particularly  to  be  remarked  about  the  time  of  the  new  and 
full  moon. 

(754.)  Quite  different  in  its  origin  is  that  deviation  of  tha  time  of  high 
and  low  water  at  any  port  or  harbour,  from  the  culminition  of  the  lumi- 
naries, or  of  the  theoretical  maximum  of  their  superposcc.  spheroids,  which 
is  called  the  "establishment"  of  that  port.  If  the  water  were  without 
inertia,  and  free  from  obstruction,  either  owing  to  the  friction  of  the  bod 
of  the  sea,  the  narrowness  of  channels  along  which  the  wave  has  to  travel 
before  reaching  the  port,  their  length,  &c.,  &c.,  the  times  above  distiu- 


OF  THE   TIDES. 


428 


Wj. 


guiehcd  would  be  identical,  Bnt  all  these  causes  tend  to  create  a  dif- 
ference, and  to  make  tli;U  difference  not  alilie  at  all  ports.  The  observa- 
tion of  the  establishments  of  harbours  is  a  point  of  great  maritime  im- 
portance; nor  ia  it  of  less  consequence,  theoretically  speaking,  to  a 
knowledge  of  the  true  distribution  of  the  tide-waves  over  the  globe.  In 
making  such  observations,  care  must  be  taken  not  to  confound  the  time 
of  "  slack  water,"  when  the  current  caused  by  the  tide  ceases  to  flow 
visibly  one  way  or  the  other,  and  that  of  Jiujh  or  low  water,  when  the 
level  of  the  surface  ceases  to  rise  or  fall.  These  are  totally  distinct  phe- 
nomena, and  depend  on  entirely  different  causes,  though  it  is  true  they 
may  sometimes  coincide  in  point  of  time.  They  are,  it  is  feared,  too  often 
mistaken  one  for  the  other  by  practical  men  j  a  circumstance  which, 
whenever  it  occurs,  must  produce  the  greatest  confusion  in  any  attempt  to 
reduce  the  system  of  the  tides  to  distinct  and  intelligible  laws. 

(755.)  The  declination  of  the  sua  and  moon  materially  affects  the  tides 
at  any  particular  spot.  As  the  vertex  of  the  tide-wave  tends  to  place 
itself  vertically  under  the  luminary  which  produces  it,  when  this  vertical 
changes  its  point  of  incidence  on  the  surface,  the  tide-wave  must  tend  <o 
shift  accordingly,  and  thus,  by  monthly  and  annual  periods,  must  tend  to 
increase  and  diminish  alternately  the  principal  tides.  The  period  of  the 
moon's  nodes  is  thus  introduced  into  this  subject;  her  excursions  in  de- 
clination in  one  part  of  that  period  being  29°,  and  in  another  only  17°, 
on  either  side  the  equator. 

(756.)  Geometry  demonstrates  that  the  efficacy  of  a  luminary  in  raising 
tides  is  inversely  proportional  to  the  cube  of  its  distance.  The  sun  and 
moon,  however,  by  reason  of  the  ellipticity  of  their  orbits,  are  alternately 
nearer  to  and  farther  from  the  earth  than  their  mean  distances.  In  eon- 
sequence  of  this,  the  efficacy  of  the  sun  will  fluctuate  between  the  ex- 
tremes 19  and  21,  taking  20  for  its  mean  value,  and  that  of  the  moon 
between  43  and  59.  Taking  into  account  this  cause  of  difference,  the 
highest  spring  tide  will  be  to  the  lowest  neap  as  59  +  21  to  43  — 19,  or 
as  80  to  24,  or  10  to  8.  Of  all  the  causes  of  differences  in  the  height 
of  tides  however,  local  situation  is  the  most  influential.  In  some  places 
the  tide-wave,  rushing  up  a  narrow  channel,  is  suddenly  raised  to  an  ex- 
traordinary height.  At  Annapolis,  for  instance,  in  the  Bay  of  Fundy,  it 
is  said  to  rise  120  feet.  Even  at  Bristol  the  difference  of  high  and  low 
water  occasionally  amounts  to  50  feet. 

(757.)  It  is  by  means  of  the  perturbations  of  the  planets,  as  ascertained 
by  observatiou  and  compared  with  theory,  that  we  arrive  at  a  knowledge 
of  the  masses  of  those  planets  which  having  no  satellites,  offer  no  other 
hold  upon  them  for  this  purpose.     Every  planet  produces  an  amount  of 


'     1 


t  i 


-nil 


424 


OUTLINES  OP  ASTRONOMY. 


perturbation  in  tbo  motions  of  every  other,  proportioned  to  its  mass,  and 
to  the  di'groe  of  advantage  or  purchase  which  ita  situation  in  the  nystctu 
gives  it  over  their  movements.  The  latter  is  a  subject  of  exact  ciilculu- 
tion ;  the  former  is  unknown,  otherwise  than  by  observation  of  its  effects. 
In  the  determination,  however,  of  the  masses  of  the  planets  by  this 
means,  theory  lends  the  greatest  assistance  to  observation,  by  pointing  out 
the  combinations  most  favourable  for  eliciting  this  knowledge  from  the 
confused  mass  of  superposed  inequalities  which  affect  every  observed  place 
of  a  planet ;  by  pointing  out  the  laws  of  each  inequality  in  its  poriodicnl 
rise  and  decay ;  and  by  showing  how  every  particular  inequality  depends 
for  its  magnitude  on  the  mass  producing  it.  It  is  thus  that  the  mass  of 
Jupiter  itself  (employed  by  Laplace  in  his  investigations,  and  interwoven 
with  all  the  planetary  tables)  has  been  ascertained,  by  observations  of  the 
derangements  produced  by  it  in  the  motions  of  the  ultra-zodiacal  planets, 
to  have  been  insufficiently  determined,  or  rather  considerably  mistaken, 
by  relying  too  much  on  observations  of  its  satellites,  made  long  ago  by 
Pound  and  others,  with  inadequate  instrumental  means.  The  same  con- 
elusion  has  been  arrived  at,  and  nearly  the  same  mass  obtained,  by  means 
of  the  perturbations  produced  by  Jupiter  on  Encke's  comet.  The  error 
was  one  of  great  importance ;  the  mass  of  Jupiter  being  by  fur  the  most 
influential  element  in  the  planetary  system,  after  that  of  the  sun.  It  is 
satisfactory,  then,  to  have  ascertained,  as  Mr.  Airy  has  done,  the  cause  of 
the  error ;  to  have  traced  it  up  to  its  source,  in  insufficient  micrometric 
measurements  of  the  greatest  elongations  of  the  satellites;  and  to  have 
found  it  disappear  when  measures,  taken  with  more  care  and  with  infinitely 
superior  instruments,  are  substituted  for  those  before  employed. 

(758.)  In  the  same  way  that  the  perturbations  of  the  planets  luad  us 
to  a  knowledge  of  their  masses,  as  compared  with  that  of  the  sun,  so  the 
perturbations  of  the  satellites  of  Jupiter  have  led,  and  those  of  Saturn's 
attendants  will  no  doubt  hereafter  lead,  to  a  knowledge  of  the  proportion 
their  masses  l^far  to  their  res^ctive  primaries.  The  system  of  Jupiter's 
sateL.es  hiis  been  elaboraiely  treated  by  Laplace;  and  it  is  from  his 
theory,  «(MA|Mtf«d  with  innumerable  observations  of  their  eclipses,  that  the 
maaaM  Magwd  tio  them,  in  art.  540  have  been  fiixed.  Few  results  of 
theory  are  iBore  surprising  than  to  sec  these  minute  atoms  weighed  in  the 
same  balance,  which  we  have  applied  to  the  ponderous  mass  of  the  sun, 
which  exceeds  the  least  of  them  in  the  enormous  proportion  of  65,000,000 

(759  )  The  mass  of  the  moon  is  concluded,  1st,  frona  the  proportion  of 
the  lunar  to  the  solar  tide,  as  observed  at  various  stations,  the  effects  being 
separated  from  each  cither  by  a  long  series  of  observations  of  the  relative 


MASS  OF  THE   MOON  DISCOVERED. 


42/) 


?n!) 


heights  of  spring  and  noap  tides  which,  wo  havo  seen,  (art.  752,)  depends 
on  the  proportional  influonoo  of  the  two  luminaries.  'Jdly,  from  the 
plioonomenon  of  nutation,  which,  being  the  result  of  the  moon's  attraction 
alone,  affords  a  means  of  calculating  her  mass,  independent  of  any  know- 
ledge of  the  sun's.  Both  methods  agree  in  assigning  to  our  satellite  a 
mass  about  one  seventy-fifth  that  of  the  earth.' 

(7G0.)  Not  only,  however,  has  a  knowledge  of  the  perturbations  pro- 
duced on  other  bodies  of  our  system  enabled  us  to  estimate  the  mass  of  a 
diflturbiug  body  already  known  to  exist,  and  to  produce  dit<turbnnce.  It 
has  done  much  more,  and  enabled  geometers  to  satisfy  thoiiisclves  of  the 
existence,  and  even  to  indicate  the  situation  ot  a  planet  previously  un- 
known, with  such  precision,  as  to  lead  to  its  immediate  diHcovery  on  the 
very  first  occasion  of  pointing  a  telescope  to  the  place  indicated.  We 
have  already  (art.  506,)  bad  occasion  to  mention  in  general  terms  this 
great  discovery  j  but  its  importance,  and  its  connexion  with  the  subject 
before  us,  calls  for  a  more  specific  notice  of  the  circumstances  attending  it. 
When  the  regular  observation  of  Uranus,  consequent  on  its  discovery  in 
1781,  had  afforded  some  certain  knowledge  of  the  elements  of  its  orbit,  it 
became  possible  to  calculate  backwards  into  time  past,  with  a  view  to 
ascertain  whether  certain  stars  of  about  the  same  apparent  magnitude, 
obiscrvcd  by  Flamsteed,  and  since  reported  as  missing,  might  not  possibly 
be  this  planet.  No  less  than  six  ancient  observations  of  it  as  a  supposed 
star  were  thus  found  to  have  been  recorded  by  that  astronomer,  —  one  in 
1690,  one  in  1712,  and  four  in  1715.  On  further  inquiry,  it  was  also 
ascertainod  to  have  been  observed  by  Bradley  in  1753,  by  Mayer  in  1756, 
and  no  \o^>  than  twelve  times  by  Le  Monnier,  in  1750,  1764, 1768, 1769, 
and  "1  ill  the  time  without  the  least  suspicion  of  its  planetary  nature. 
Thf  .^bstTvations,  however,  so  made,  being  all  circumstantially  registered, 
ttst  made  with  instruments  the  best  that  their  respective  dates  admitted, 
"were  quite  available  for  correcting  the  elements  of  the  orbit,  which,  as 
will  be  easily  understood,  is  done  with  so  much  the  greater  precision  the 
larger  the  arc  of  the  ellipse  embraced  by  the  extreme  observations  em- 
ployed. It  was,  therefore,  reasonably  hoped  and  expected,  that,  by 
making  use  of  the  data  thus  afforded,  and  duly  allowing  for  the  perturba- 
tions produced  since  1690,  by  Saturn,  Jupiter,  and  the  inferior  planets, 
elliptic  elements  would  be  obtained,  which,  taken  in  conjunction  with 
those  perturbations,  would  represent  not  only  all  the  observations  up  to 
the  time  of  executing  the  calculations,  but  also  all  future  observations,  in 
as  satisfactory  a  manner  as  those  of  any  of-  the  other  planets  are  actually 
represented.     This  expectation,  however,  proved  delusive.     M.  Bouvard, 

» Laplace,  Expos,  du  Syst.  du  Monde,  pp.  285,  300. 


'    I 


a 
* 


426 


OUTLINES   OF  ASTRONOMY. 


one  of  the  most  expert  aud  laborious  calculators  of  whom  astronomy  has 
had  to  boast,  and  to  whose  zeal  and  indefatigable  industry  we  owe  the 
tables  of  Jupiter  and  Saturn  in  actual  use,  having  undertaken  the  task  of 
constructing  similar  tables  for  Uranus,  found  it  impossible  to  reconcile 
the  ancient  observations  above  mentioned  with  those  made  from  1781  to 
1820,  so  as  to  represent  both  series  by  means  of  the  same  ellipse  and  the 
same  system  of  perturbations.  He  therefore  rejected  altogether  the  ancient 
series,  and  grounded  his  computations  solely  on  the  modem,  although 
evidently  not  without  serious  misgivings  as  to  the  grounds  of  such  a  pro- 
ceeding, and  "  leaving  it  to  future  time  to  determine  whether  the  difficulty 
of  reconciling  the  two  series  arose  from  inaccuracy  in  the  older  observa- 
tions, or  whether  it  depend  on  some  extraneous  and  unperceived  influence 
which  may  have  acted  on  the  planet." 

(761.)  But  neither  did  the  tables  so  calculated  continue  to  represent, 
with  due  precision,  observations  subsequently  made.  The  "  error  of  the 
tables"  after  attaining  a  certain  amount,  by  which  the  true  longitude  of 
Uranus  was  in  advance  of  the  computed,  and  which  advance  was  steadily 
maintained  from  about  the  year  1795  to  1822,  began,  about  the  latter 
epoch,  rapidly  to  diminish,  till,  in  1830-31,  the  tabular  and  observed 
longitudes  agreed.  But,  far  from  remaining  in  accordance,  the  planet, 
still  losing  ground,  fell,  and  continued  to  fall  behind  its  calculated  place, 
and  that  with  such  rapidity  as  to  make  it  evident  that  the  existing  tables 
could  no  longer  be  received  as  representing,  with  any  tolerable  precision, 
the  true  laws  of  its  motion. 

(762.)  The  reader  will  easily  understand  the  nature  and  progression  of 
these  discordancies  by  casting  his  eye  on  fig.  1,  Plate  A,  in  which  the 
horizontal  line  or  abscissa  is  divided  into  equal  parts,  each  representing 
50**  of  heliocentric  longitude  in  the  motion  of  Uranus  round  the  sun,  and 
in  which  the  distances  between  the  horizontal  lines  represent  each  100" 
of  error  in  longitude.  The  result  of  each  year's  observation  of  Uranus 
(or  of  the  mean  of  all  the  observations  obtained  during  that  year)  in  lon- 
gitude, is  represented  by  a  black  dot  placed  above  or  below  the  point  of 
the  abscissa,  corresponding  to  the  mean  of  the  observed  longitudes  for  the 
year  above,  if  the  observed  longitude  be  in  excess  of  the  calculated,  below 
if  it  fall  short  of  it,  and  on  the  line  if  they  agree  j  and  at  a  distance  from 
the  line  corresponding  to  their  difference  on  the  scale  above  mentioned.' 

*  The  points  are  laid  down  from  M.  Leverrier's  comparison  of  the  whole  series  of 
observations  of  Uranus,  with  an  ephemeris  of  his  own  calculation,  founded  on  a  com- 
plete and  searching  revision  of  the  tables  of  Bouvard,  and  a  rigorous  computation  of 
the  perturbations  caused  by  all  the  known  planets  capable  of  exercising  any  influence 
on  it.  The  diffi  rences  of  longitude  are  geocentric;  but  for  our  present  purpose  it 
matters  not  ip  the  least  whether  we  consider  the  errors  in  heliocentric  or  in  geocentric 
longitude. 


PERTURBATION   OF   URANUS. 


427 


Thus  in  Flamsteed's  earliest  observations  in  1690,  the  dot  so  marked  is 
placed  above  the  line  at  65" -9  above  the  line,  the  observed  longitude  being 
BO  much  greater  than  the  calculated. 

(763.)  If,  neglecting  the  individual  points,  we  draw  a  curve  (indicated 
in  the  figure  by  a  fine,  unbroken  line,)  through  their  general  course,  we 
shall  at  once  perceive  a  certain  regularity  in  its  undulations.  It  presents 
two  great  elevations  above,  and  one  nearly  as  great  intermediate  depres- 
sion below  the  medial  line  or  abscissa.  And  it  is  evident  that  these  un- 
dulations would  be  very  much  reduced,  and  the  errors  in  consequence 
greatly  palliated,  if  each  dot  were  removed  in  the  vertical  direction 
through  a  distance  and  in  the  direction  indicated  by  the  corresponding 
point  of  the  curve  A,  B,  G,  D,  E,  F,  G,  H,  intersecting  the  abscissa  at 
points  180°  distant,  and  making  equal  excursions  on  either  side.  Thus 
the  point  a  for  1750  being  removed  upwards  or  in  the  direction  to- 
wards h  through  a  distance  equal  to  ch,  would  be  brought  almost  to 
precise  coincidence  with  the  point  e  in  the  abscissa.  Now,  this  is  a 
clear  indication  that  a  very  large  part  of  the  differences  in  question  are 
due,  not  to  perturbation,  but  simply  to  error  in  the  elements  of  Uranus, 
which  have  been  assumed  as  the  basis  of  calculation..  For  such  ex- 
cesses and  defects  of  longitude  alternating  over  arcs  of  180°  are  pre- 
cisely what  would  arise  from  error  in  the  excentricity,  or  in  the  place 
of  the  perihelion,  or  in  both.  In  ellipses  slightly  escentric,  the  true  lon- 
gitude alternately  exceeds  and  fulls  short  of  the  mean  during  180°  for 
each  deviation,  and  the  greater  the  excentricity,  the  greater  these  alternate 
fluctuations  to  and  fro.  If  then  the  excentricity  of  a  planet's  orbit  bo 
assumed  erroneously  (suppose  too  great)  the  observed  longitudes  will  ex- 
hibit a  less  amount  of  such  fluctuation  above  and  below  the  mean  than 
the  computed,  and  the  difference  of  the  two,  instead  of  being,  as  it  ought 
to  be,  always  nil,  will  be  alternately  +  and  —  over  arcs  of  180°.  If  then 
a  difference  be  observed  following  such  a  law,  it  may  arise  from  erro- 
neously assumed  excentricity,  provided  always  the  longitudes  at  which 
they  agree  (supposed  to  differ  by  180°)  be  coincident  with  those  of  the 
perihelion  and  aphelion ;  for  in  elliptic  motion  nearly  circular,  these  are 
the  points  where  the  mean  and  true  longitudes  agree,  so  that  any  fluctua- 
tion of  the  nature  observed,  if  this  condition  be  not  satisfieu,  cannot  arise 
from  error  of  excentricity.  Now  the  longitude  of  the  perihelion  of 
Uranus  in  the  elements  employed  by  Bouvard  is  (neglecting  fractions  ot 
a  degree)  168°,  and  of  the  aphelion  348°.  These  points,  then,  in  our 
figure,  fall  at  h  and  a,  respectively,  that  is  to  say,  nearly  half  way  between 
A  C,  C  E,  p]  Gr,  &c.  It  is  evident,  therefore,  that  it  is  not  to  error  or 
excentricity  that  the  fluctuation  in  question  is  mainly  due. 


:-!iiS 


;  «;i:i, 


428 


OUTLINES   OP  ASTRONOMY. 


(764.)  Let  us  now  consider  the  effect  of  an  erroneous  assumption  of 
the  place  of  the  perihelion.  Suppose  in  fig.  2,  Plate  A,  oo;  to  represent 
the  longitude  of  a  planet,  and  xy  the  excess  of  its  true  above  its  mean 
longitude,  due  to  ellipticity.  Then  if  R  be  the  place  of  the  perihelion, 
and  P,  or  T,  the  aphelion  in  longitude,  y  will  always  lie  in  a  certain  un- 
dulating curve  P  Q  R  S  T,  above'  P  T,  between  R  and  T,  and  below  it 
between  P  and  R.  Now  suppose  the  place  of  the  perihelion  shifted  for- 
ward to  r,  or  the  whole  curve  shifted  bodily  forward  into  the  situation 
pqrst,  then  at  the  same  longitude  o x,  the  excess  of  the  true  above  the 
mean  longitude  will  be  x^  onlyj  in  other  words,  this  excess  will  have 
diminished  by  the  quantity  yy'  below  its  former  amount.  Take  therefore 
in  oN  (^(/.  3,  PI.  A,)  oy  =  ox  and  yi/  always  =^yj/  in  jig.  2,  and 
having  thus  constructed  the  curve  KLMNO,  the  ordinate  yt/  will 
always  represent  the  effect  of  the  supposed  change  of  perihelion.  It  is 
evident  (the  excentricity  being  always  supposed  small),  that  this  curve 
will  consist  also  of  alternate  superior  and  inferior  waves  of  180°  each  in 
amplitude,  and  the  points  L,  N  of  its  intersection  with  the  axis,  will 
occur  at  longitudes  corresponding  to  X,  Y,  intermediate  between  th' 
maxima  Q,  q ;  and  S,  .s  of  the  original  curves,  that  is  to  say  (if  these  m 
tervals  Q  g-,  S  s,  or  R  r,  to  which  both  are  equal,  be  very  small,)  ver^ 
nearly  at  90°  from  the  perihelion  and  aphelion.  Now  this  agrees  with 
the  conditions  of  the  case  in  hand,  and  we  are  therefore  authorised  to 
conclude  that  the  major  portion  of  the  errors  in  question  has  arisen  from 
error  in  the  place  of  the  perihelion  of  Uranus  itself,  and  not  from  pertur- 
bation, and  that  to  correct  this  portion,  the  perihelion  must  be  shifted 
somewhat  forward.  As  to  the  amount  of  this  shifting,  our  only  object 
being  explanation,  it  will  not  be  necessary  here  to  inquire  into  it.  It  will 
suflSce  that  it  must  be  such  as  shall  make  the  curve  ABCDEFG-  as 
nearly  as  possible  similar,  equal,  and  opposite  to  the  curve  traced  out  by 
the  dots  on  the  other  side.  And  this  being  done,  we  may  next  proceed 
to  lay  down  a  curve  of  the  residual  differences  between  observation  and 
theory  in  the  mode  indicated  in  art.  (763.) 

(765.)  This  being  done,  by  laying  off  at  each  point  of  the  line  of 
longitudes  an  ordinate  equal  to  the  difference  of  the  ordinates  of  the  two 
curves  in  fig.  1,  when  on  opposite,  and  their  sum  when  on  the  same  side 
of  the  abscissa,  the  result  will  be  as  indicated  by  the  dots  in  fig.  4.  And 
here  it  is  at  once  seen  that  a  still  farther  reduction  of  the  differences  under 
consideration  would  result,  if,  instead  of  taking  the  line  A  B  for  the  line 
of  longitudes,  a  line,  a  b,  slightly  inclined  to  it,  were  substituted,  in  which 
case  the  whole  of  the  differences  between  observation  and  theory,  from 


fe 


The  curves,  figs,  2,  3,  are  inverted  in  the  engraving. 


0^ 


PERTURBATIONS   OF  URANUS  BY  NEPTUNE. 


429 


1712  to  1800,  would  be  annihilated,  or  at  least  so  far  reduced  as  hardly 
to  exceed  the  ordinary  errors  of  observation ;  and  as  respects  the  observa- 
tion of  1690,  the  still  outstanding  difference  of  about  85"  would  not  be 
more  than  might  be  attributed  to  a  not  very  careful  observation  at  so  early 
an  epoch.  Now  the  assumption  of  such  a  new  line  of  longitudes  as  the 
correct  one,  is  in  effect  equivalent  to  the  admission  of  a  slight  amount  of 
error  in  the  periodic  time  and  epoch  of  Uranus ;  for  it  is  evident  that  by 
reckoning  from  the  inclined  instead  of  the  horizontal  line,  we  in  effect 
alter  all  the  apparent  outstanding  errors  by  an  amount  proportional  to  the 
time  before  or  after  the  date  at  which  the  two  lines  intersect  (viz.  about 
1789).  As  ut  the  direction  in  which  this  correction  should  be  made,  it  is 
obvious  by  inspection  of  the  course  of  the  dots,  that  if  we  reckon  from 
A  B,  or  any  line  parallel  to  it,  the  observed  planet  on  the  long  run  keeps 
falling  more  and  more  behind  the  calculated  one ;  i.  e.  its  assigned  mean 
angular  velocity  by  the  tables  is  too  great,  and  must  be  diminished,  or  its 
periodic  time  requires  to  be  increased. 

(766.)  Let  this  increase  of  period  be  made,  and  in  correspondence  with 
that  change  let  the  longitudes  be  reckoned  on  a  6,  and  the  residual  diffe- 
rences from  that  line  instead  of  A  B,  and  we  shall  have  then  done  all  that 
can  be  done  in  the  way  of  reducing  and  palliating  these  differences,  and 
that,  with  such  success,  that  up  to  the  year  1804  it  might  have  been  safely 
asserted  that  positively  no  ground  whatever  existed  for  suspecting  any 
disturbing  influence.  But  with  this  epoch  an  action  appears  to  have 
commenced,  and  gone  on  increasing,  producing  an  acceleration  of  the 
motion  in  longitude,  in  consequence  of  which  Uranus  continually  gains  on 
its  elliptic  place,  and  continued  to  do  so  till  1822,  when  it  ceased  to  gain, 
and  the  excess  of  longitude  was  at  its  maximum,  after  which  it  began 
rapidly  to  lose  ground,  and  has  continued  to  do  so  up  to  the  present  time. 
It  is  perfectly  clear,  then,  that  in  this  interval  some  extraneous  cause  must 
have  come  into  action  which  was  not  so  before,  or  not  in  sufficient  power 
to  manifest  itself  by  any  marked  effect,  and  that  that  cause  must  have 
ceased  to  act,  or  rather  begun  to  reverse  its  action,  in  or  about  the  year 
1822,  the  reverse  action  being  even  more  energetic  than  the  direct. 

(767.)  Such  is  the  phaenomenon  in  the  simplest  form  we  are  now  able 
to  present  it.  Of  the  various  hypotheses  formed  to  account  for  it,  during 
the  progress  of  its  developement,  none  seemed  to  have  any  degree  of  ra- 
tional probability  but  that  of  the  existence  of  an  exterior,  and  hitherto 
undiscovered,  planet,  disturbing,  according  to  the  received  laws  of  plan- 
etary disturbance,  the  motion  of  Uranus  by  its  attraction,  or  rather  super- 
posing its  disturbance  on  those  produced  by  Jupiter  and  Saturn,  the  only 
two  of  the  old  planets  which  exercise  any  sensible  disturbing  action  on 


*!;!<•: 


I  r  k. 


430 


OUTLINES  OF  ASTRONOMY. 


that  planet.  Accordingly,  this  was  the  explanation  which  naturally,  and 
almost  of  necessity,  suggested  itself  to  those  conversant  with  the  plane* 
tary  perturbations  who  considered  the  subject  with  any  degree  of  atten- 
tion. The  idea,  however,  of  setting  out  from  the  observed  anomalous 
deviations,  and  employing  them  as  data  to  ascertain  the  distance  and  situ- 
ation of  the  unknown  body,  or,  in  other  words,  to  resolve  the  inverse 
problem  of  perturbations,  "given,  the  disturbances  to  find  the  orbit,  and 
place  in  that  orbit  of  the  disturbing  planet,"  appears  to  have  occurred 
only  to  two  mathematicians,  Mr.  Adams  in  Er.i;land  and  M.  le^errier  in 
France,  with  sufficient  distinctness  and  hopefulness  of  success  to  induce  them 
to  attempt  its  solution.  Both  succeeded,  and  their  i^olutions,  arrived  at  with 
perfect  independence,  and  by  each  in  entire  ignorance  of  the  other's  atr 
tempt,  were  found  to  agree  in  a  surpiising  manner  when  the  nature  and 
difiEculty  of  the  problem  is  considered ;  the  calculations  of  M.  Lovcrrier 
assigning  for  the  heliocentric  longitude  of  the  disturbing  planet  for  the 
23d  Sept.  1846,  326°  0',  and  those  of  Mr.  Adamfi  (brought  to  the  same 
date)  329°  19',  diflPerihg  only  3"  19' ;  the  plane  of  its  orbit  deviating  very 
slightly,  if  it  all,  from  that  of  the  ecliptic. 

(768.)  On  the  day  above  mentioned  —  a  day  for  ever  memorable  in  the 
annals  of  astronomy — Dr.  Galle,  one  of  the  astronomers  of  the  Royal 
Observatory  at  Berlin,  received  a  letter  from  M.  Leverrier,  announcing  to 
him  the  result  he  had  arrived  at,  and  requesting  him  to  look  for  the  dis- 
turbing planet  in  or  near  the  place  assigned  by  his  calculation.  He  did 
so,  and  on  that  very  night  actuallij  found  it.  A  star  of  the  eighth  mag- 
nitude was  seen  by  him  and  by  M.  Encke  in  a  situation  where  no  star 
was  marked  as  existing  in  Dr.  Bremiker's  chart,  then  recently  published 
by  the  Berlin  Academy.  The  next  night  it  was  found  to  have  moved 
from  its  place,  and  was  therefore  assuredly  a  planet.  Subsequent  oliser- 
vations  and  calculations  have  fully  demonstrated  this  planet,  to  which  the 
name  of  Neptune  has  been  assigned,  to  be  really  that  body  to  whose  dis- 
turbing attraction,  according  to  the  Newtonian  law  of  gravity,  the  observed 
anc^malies  in  the  motion  of  Uranus  were  owing.  The  geocentric  longi- 
tude determined  by  Dr.  Galle  from  this  observation  was  325°  53',  which, 
converted  into  heliocentric,  gives  326°  52',  differing  0°  52'  froia  M.  Le- 
verrier's  place,  2°  27'  from  that  of  Mr.  Adams,  and  only  47'  from  a 
mean  of  the  two  calculations. 

(769.)  It  would  be  quite  beyond  the  scope  of  this  work,  and  far  in 
advance  of  the  amount  of  mathematical  knowledge  we  have  assumed  our 
readers  to  possess,  to  attempt  giving  more  than  a  superficial  idea  of  the 
course  followed  by  these  geometers  in  their  arduous  investigations.  Suf- 
fice it  to  say,  that  it  consisted  in  regarding,  as  unknown  quantities,  to  be 


PERTURBATI  NS  OP  URANUS  BY  NEPTUNE. 


431 


determined,  the  mass,  and  all  the  elements  of  the  unknown  planet  (sup- 
posed to  revolve  in  the  same  plane  and  the  same  direction  with  Uranus), 
except  its  major  semiaxis.  This  was  assumed  in  the  first  instance  (in 
conformity  with  "  Bode's  law,"  art.  (505),  and  certainly  at  the  time  with 
a  high  primd  facie  r-.i^Lability,)  to  be  double  that  of  Uranus,  or  38-36't 
radii  of  the  Earth's  orbit.  Without  some  assumption  as  to  the  value  of 
this  element,  owing  to  the  peculiar  form  of  the  analytical  expression  of 
the  perturbations,  the  analytical  investigation  would  have  presented  difl5- 
culties  apparently  insuperable.  But  besides  these,  it  was  also  necessary 
to  regard  as  unknown,  or  at  least  as  liable  to  corrections  of  unknown 
magnitude  of  the  same  order  as  the  perturbations,  all  the  elements  of 
Uranus  itself,  a  circumstance  whose  necessity  will  be  easily  understood, 
when  we  consider  that  the  received  elements  could  only  be  regarded  as 
provisional,  and  must  certainly  be  erroneous,  the  places  from  which  they 
were  obtained  being  affected  by  at  least  some  portions  of  the  very  pertur- 
bations in  question.  This  consideration,  though  indispensable,  added 
vastly  both  to  the  complication  and  the  labour  of  the  inquiry.  The  axis 
(and  therefore  the  mean  motion)  of  the  one  orbit,  then,  being  known  very 
nearly,  and  that  of  the  other  thus  hypothetically  assumed,  it  became  prac* 
ticable  to  express  in  terms,  partly  algebraic,  partly  numerical,  the  amount 
of  perturbation  at  any  instant,  by  the  aid  of  general  expressions  delivered 
by  Laplace  in  his  "  Micanique  Cileste"  and  elsewhere.  These,  then, 
together  with  the  corrections  due  to  the  altered  elements  of  Uranus  itself, 
being  applied  to  the  tabular  longitudes,  furnished,  when  compared  with 
those  observed,  a  series  of  equations,  in  which  the  elements  and  mass  of 
Neptune,  and  the  corrections  of  those  of  Uranus  entered  as  the  unknown 
quantities,  and  by  whose  resolution  (no  slight  effort  of  analytical  skill)  all 
their  values  were  at  length  obtained.  The  calculations  were  then  repeated, 
reducing  at  the  same  time  the  value  of  the  assumed  distance  of  the  new 
planet,  the  discordances  between  the  given  and  calculated  results  indicating 
it  to  have  been  assumed  too  large  when  the  results  were  found  to  agree 
better,  and  the  solutions  to  be,  in  fact,  more  satisfactory.  Thus,  at  length, 
elements  were  arrived  at  for  the  orbit  of  the  unknown  planet,  as  below. 


Leverrier. 

Adams. 

Epoch  of  Elements 

Jan.  1, 1847. 

318°  47'  4" 
36-1539 
0-107610 

284°  45'  8" 

0-00010727 

Oct.  6,  1846. 
323°  2' 
37-2474 
0-120615 
299°  11' 
0-00015003 

Mean  longitude  in  Epoch 

Semiaxis  Maior 

Excmtrioity 

Longitude  of  Perihelion 

Mass  (the  Sun  being  1) 

'1 


432 


OUTLINES  OF  ASTRONOMY. 


Tbo  elenicuts  of  M.  Leverrier  were  obtained  from  a  consideration  of  the 
observations  up  to  the  year  1845,  those  of  Mr.  Adams  only  as  far  aa 
1840.  On  subsequently  taking  into  account,  however,  those  of  the  fivo 
years  up  to  1845,  the  latter  was  led  to  conclude  that  the  semiazis  ought 
to  be  still  much  further  diminished,  and  that  a  mean  distance  of  33-33 
(being  to  that  of  Uranus  as  1 :  0-574)  would  probably  satisfy  all  the  obser- 
vations very  nearly.' 

(770.)  On  the  actual  discovery  of  the  planet,  it  was,  of  course,  assidu- 
ously observed,  and  it  was  soon  ascertained  that  a  mean  distance,  even  less 
than  Mr.  Adams's  last  presumed  value,  agreed  better  with  its  motion ; 
and  no  sooner  were  elements  obtained  from  direct  observation,  sufficiently 
approximate  to  trace  back  its  path  in  the  heavens  for  a  considerable  inter- 
val of  time,  than  it  was  ascertained  to  have  been  observed  as  a  star  by 
Lalande  on  the  8th  and  10th  of  May,  1795,  the  latter  of  the  two  obser- 
vations, however,  having  been  rejected  by  him  as  faulty,  by  reason  of  its 
non-agreement  with  the  former  (a  consequence  of  the  motion  of  the 
planet  in  the  interval.)  From  these  observations,  combined  with  those 
since  accumulated,  the  elements  calculated  by  Prof.  Walker,  U.  S.,  result 
as  follows : — 


Epoch  of  Elements 

Mean  longitude  at  Epoch 

Semiaxis  major 

Excentricity 

Longitude  of  the  Perihelion 

Ascending  Node 

Inclination 

Periodic  time 

Mean  annual  Motion 


Jan.  1,  1847,  M.  noon,  Greenwich. 
328°  32'  44"  2 
30-0367 

000871946 

47°13'6"-50 

130°  4'  20"-81 

l°46'58"-97 
164-6181  tropical  year. 

2°-18688 


I 


(771.)  The  great  disagreement  between  those  elements  and  t.hosc  assigned 
either  by  M.  Leverrier  or  Mr.  Adams  will  not  fail  to  be  remarked ;  and  it 
will  naturally  be  asked  how  it  has  come  to  pass,  that  elements  so  widely 
different  from  the  truth  should  afford  anything  like  a  satisfactory  represen- 
tation of  tlio  perturbation  in  question,  and  that  the  true  situation  of  the 
planet  in  the  heavens  should  have  been  so  well,  and  indeed  accurately, 
pointed  out  by  them.  As  to  the  latter  point,  any  one  may  satisfy  himself 
by  half  an  hour's  calculation  that  both  sets  of  elements  do  really  place  the 
planet,  on  the  day  of  its  discovery,  not  only  in  the  longitudes  assigned  in 
art.  763,  i.  e.  extremely  near  its  apparent  place,  but  also  at  a  distance  from 
the  sun  very  much  more  approximately  correct  than  the  mean  distances  or 
semiaxes  of  the  respective  orbits.     Thus  the  radius  vector  of  Neptune, 


'  In  a  letter  to  the  Astronomer  Royal,  dated  Sept.  2, 1846,— t.  e.  three  weeks  previous 
to  the  optical  discovery  of  the  planet. 


PERTURBATIONS  OF  URANUS  BY  NEPTUNE. 


483 


calculated  from  M.  Leverrier's  eleraonta  for  the  day  in  question,  instead 
of  36-1539  (the  mean  distance)  comes  out  almost  exactly  33;  and  indeed, 
if  wo  consider  that  the  excentricity  assigned  by  those  elements  gives  for 
the  perihelion  distance  32-2634,  the  longitude  assigned  to  the  perihelion 
brings  the  whole  arc  of  the  orbit  (more  than  83®).  described  in  the  in- 
terval from  1806  to  1847  to  lie  within  42°  one  way  or  the  other  of  the 
perihelion,  and  therefore,  during,  the  whole  of  that  interval,  the  hypothe- 
tiaal  planet  would  be  moving  within  limits  of  distance  from  the  sun,  32-6 
and  33-0.  The  following  comparative  tables  of  the  relative  situations  of 
Uranus,  the  real  and  hypothetical  planet,  will  exhibit  more  clearly  than 
any  lengthened  statement,  the  near  imitation  of  the  motion  of  the  former 
by  the  latter  within  that  interval.     The  longitudes  are  heliocentric' 


'\J^\ 


f  T3 


Uranui. 

Neptu 

ne. 

Leverrier. 

Adams. 

A.D. 

Long. 

Long. 

Rad.  Veo. 

Long. 

Rod.  Veo. 

Long. 

Rad.  Vec. 

1805-0 

197°-8 

235°-9 

30-3 

241°-2 

33-1 

246°-6 

34-2 

1810-0 

220-9 

247-0 

30-3 

251-1 

32-8 

255-9 

33-7 

1815-0 

243-2 

268-0 

30-3 

261-2 

32-5 

265-5 

33-3 

1820-0 

264-7 

268-8 

30-2 

271-4 

32-4 

275-4 

331 

1821-0 

209-0 

271-0 

30-2 

273-5 

32-3 

277-4 

330 

1822-0 

273-3 

273-2 

30-2 

275-6 

32-3 

279-5 

33-0 

182;i-0 

277-0 

275-3 

30-2 

277-6 

32-3 

281-5 

32-9 

1821-0 

281-» 

277-4 

30-2 

279-7 

32-3 

283-6 

32-9 

1825-0 

285-8 

279-6 

30-2 

281-8 

32-3 

285-6 

32-8 

18;i0-0 

306-1 

290-5 

30-1 

292-1 

32-3 

296-0 

32-8 

1835-0 

320-0 

301-4 

30-1 

302-5 

32-4 

306-3 

32-8 

1840-0 

345-7 

312-2 

30-1 

312-6 

32-6 

316-3 

32-9 

1845-0 

365-3 

323-1 

30-0 

322-6 

32-9 

3260 

?3-l 

1847-0 

373-3 

327-6 

30-0 

326-5 

33-1 

32'  3 

33-2 

\'^{[h 


•'  Hi' 


(772.)  From  this  comparison  it  will  be  seen  that  Uranus  arrived  at  its 
conjunction  with  Neptune  at  or  immediately  before  the  commencement  of 
1822,  with  the  calculated  planet  of  Leverrier  at  the  beginning  of  the  fol- 
lowing year  182ti,  and  with  that  of  Adams  about  the  end  of  1824,  Both 
the  theoretical  planets,  and  especially  that  of  M.  Leverrier,  therefore, 
during  the  whole  of  the  above  interval  of  time,  so  far  as  the  directions 
of  their  attractive  forces  on  Uranus  are  concerned,  would  act  nearly  on  it 
as  the  true  planet  must  have  done.  As  regards  the  intensity  of  the  rela- 
tive disturbing  forces,  if  we  estimate  these  by  the  principles  of  art.  (612) 
at  the  epochs  of  conjunction,  and  for  the  commencement  of  1805  and 


'  The  calculations  are  carried  only  to  tenths  of  degrees,  as  quite  sufficient  for  the 
objact  in  view. 

28 


434 


OUTLINES   OF  ASTRONOMY. 


1845,  we  find  for  the  respective  denominators  of  the  fractions  of  the  sun's 
attraction  on  Uranus  regarded  as  unity,  which  express  the  total  disturbing 
force,  N  S,  in  each  case,  as  below : 


1 


1805. 

Conjunction. 

1846. 

27540 

7508 

3239C 

20244 

5519 

23810 

20837 

5193 

19935 

.  ,     f  Pierce's  mass 
Neptune  with   i  „ 

(  Struve  s  maas 

Leverrier'a  theoretical  Planet,  mass 

The  masses  here  assigned  to  Neptune  are  those  respectively  deduced  by 
Prof.  Peirce  and  M.  Struve  from  observations  of  the  satellite  discovered 
by  Mr.  Lassell  made  with  the  large  telescopes  of  Fraunhofer  in  the  obser- 
vatories of  Cambridge,  U.  S.  and  Pulkova  respectively.  These  it  will  be 
perceived  diflFer  very  considerably,  as  might  reasonably  be  expected  in  tlio 
results  of  micrometr'^^al  measurements  of  such  difficulty,  and  it  is  not  pos- 
sible at  present  to  say  to  which  the  preference  ought  to  be  given.  As 
compared  with  the  mass  assigned  by  M.  Struve,  an  agreement  on  the 
whole  more  satisfactory  could  not  have  been  looked  for  within  the  interval 
innii  nliately  in  question.  > 

(773.)  Subject  then  to  this  uncertainty  as  to  the  real  mass  of  Neptune, 
the  theoretical  planet  of  Leverrier  must  be  considered  as  representing  with 
quite  as  much  fidelity  as  could  possibly  be  expected  in  a  research  of  such 
exceeding  delicacy,  the  particulars  of  its  motion  and  perturbative  action 
during  the  forty  years  elapsed  from  1805  to  1845,  an  interval  which  (as 
is  obvious  from  the  rapid  diminution  of  the  forces  on  either  side  of  the 
conjunction  indicated  by  the  numbers  here  set  down)  comprises  all  the 
most  influential  range  of  its  action.  This  will,  however,  be  placed  iu  full 
evidence  by  the  construction  of  curves  representing  the  normal  and  tan- 
gential forces  on  the  principles  laid  down  (as  far  as  the  normal  constituent 
is  concerned)  in  art  717,  one  slight  change  only  being  made,  which,  tbr 
the  purpose  in  view,  conduces  greatly  to  clearness  of  conception.  The 
force  L  8  (in  the  figure  of  that  article)  being  supposed  applied  at  P  in  the 
direction  L  8,  we  here  construct  the  curve  of  the  normal  force  by  erecting 
at  P  (Jig.  5,  Plate  A)  P  W  always  perpendicular  to  the  disturbed  orbit, 
A  P,  at  P,  measured  from  P  in  the  same  direction  that  S  lies  from  L,  and 
equal  in  length  to  L  S.  P  W  will  then  always  represent  both  the  direc- 
tion and  magnitude  of  the  normal  force  acting  at  P.  And  in  like  man- 
ner, if  we  take  always  P  Z  on  the  tangent,  to  the  disturbed  orbit  at  P, 
equal  to  N  L  of  the  former  figure,  and  measured  in  the  same  direction 
from  P  that  L  is  from  N,  P  Z  will  represent  both  in  magnitude  and  direc- 
tion the  tangential  force  acting  at  P.  Thus  will  be  traced  out  the  two 
curious  ovals  represented  in  our  figure  of  their  proper  forms  and  propor- 
tions for  the  case  in  question.    That  expressing  the  normal  force  is  formed 


PERTURBATIONS  OF  URANUS  BY  NEPTUNE. 


435 


of  four  lobes,  having  a  common  point  in  S,  viz.,  SWmXSaSnSiS  W^ 
and  that  ezpressing  the  tangential,  AZc/Bee^YAZ,  consisting  of  four 
mutually  intersecting  loops,  surrounding  and  touching  the  disturbed  orbit 
in  four  points,  AB  cd.     The  normal  force  acts  outward  over  all  that  part 
of  the  orbit,  both  in  conjunction  and  opposition,  corresponding  to  the  por- 
tions of  tho  lobes  m,  n,  exterior  to  the  disturbed  orbit,  and  inwards  in 
every  other  part.     The  figure  sets  in  a  clear  light  the  great  disproportion 
between  the  energy  of  this  force  near  the  conjunction,  and  in  any  other 
configuration  of  the  planets ;  its  exceedingly  rapid  degradation  as  P  ap- 
prouchcs  the  point  of  neutrality  (whose  situation  is  85°  5'  on  cither  side 
of  the  conjunction,  an  arc  described  synodically  by  Uranus  in  16y-72) ; 
and  the  comparatively  short  duration  and  consequent  inefficacy  to  produce 
any  great  amount  of  perturbation,  of  the  more  intense  part  of  its  inward 
action  in  the  small  portions  of  the  orbit  corresponding  to  the  lobes  a,  L, 
in  which  the  line  representing  the  inward  force  exceeds  the  radius  of  the 
circle.     It  exhibits,  too,  with  no  less  distinctness,  the  gradual  develope- 
ment,  and  rapid  degradation  and  extinction  of  the  tangential  force  from 
its  neutral  points,  c,  d,  on  either  side  up  to  the  conjunction,  where  its 
action  is  reversed,  being  accelerative  over  the  arc  d  A,  and  retardative 
over  A  c,  each  of  which  arcs  has  an  amplitude  of  71°  20',  and  is  de- 
scribed by  Uranus  synodically  in  34^00.     The  insignificance  of  the  tan- 
gential force  in  the  configurations  remote  from  conjunction  throughout  tho 
arc  r.Bd  is  also  obviously  expressed  by  the  small  comparative  develope- 
luent  of  the  loops  c,  /. 

(774.)  Let  us  now  consider  how  the  action  of  these  forces  results  in 
the  production  of  that  peculiar  character  of  perturbation  which  is  exhi- 
bited in  our  curve.  Jig.  4,  Plate  A.  It  is  at  once  evident  that  the  increase 
of  the  longitude  from  1800  to  1822,  the  cessation  of  that  increase  in 
1822,  and  its  conversion  into  a  decrease  during  the  subsequent  interval  is 
in  complete  accordance  with  the  growth,  rapid  decay,  extinction  at  conjunc- 
tion, and  subsequent  reproduction  in  a  reversed  sense  of  the  tangential 
force :  so  that  we  cannot  hesitate  in  attributing  the  greater  part  of  the 
perturbation  expressed  by  the  swell  and  subsidence  of  the  curve  between 
the  years  1800  and  1845, — all  that  part,  indeed,  which  is  symmetrical  on 
either  side  of  1822  —  to  the  action  of  the  tangential  force. 

(775.)  But  it  will  be  asked,  —  has  then  the  normal  force  (which,  on 
the  plain  showing  of  Jig.  5,  is  nearly  twice  as  powerful  as  the  tangential, 
and  which  does  not  reverse  its  action,  like  the  latter  force,  at  the  point  of 
junction,  but,  on  the  contrary,  is  there  most  energetic,)  no  influence  in 
producing  the  observed  effects  ?  We  answer,  very  little,  within  the  period 
to  which  observation  had  extended  up  to  1845.     The  effect  of  the  tau- 


Hi 

V,Ll 


430 


OUTLINES   OP  ASTRONOMY. 


gential  force  od  the  longitude  is  direct  and  immediate  (art.  660),  that  of 
tlie  normal  indirect,  consequential,  and  cumulative  with  the  progress  of 
time  (art.  784.)  The  effect  or  the  tangential  force  on  the  mean  motion 
takes  place  through  the  medium  of  the  change  it  produces  on  the  axis, 
and  is  transient:  the  reversed  action  after  conjunction  (supposing  tho 
orbits  circular),  exactly  destroying  all  the  previous  effect,  and  leaving  the 
mean  motion  on  the  whole  unaffe.icd.  In  tho  passage  through  the  con- 
junction, then,  the  tangential  force  produces  a  sudden  and  powerful  accel- 
eration, succeeded  by  an  equally  powerful  and  equally  sudden  retardation, 
which  done,  its  action  is  completed,  and  no  trace  remains  in  the  subse- 
quent motion  of  the  planet  that  it  ever  existed,  for  its  action  on  the  peri- 
helion and  exceiitricity  is  in  like  manner  also  nullified  by  its  reversal  of 
direction.  But  with  the  normal  force  the  case  is  far  otherwise.  Its  im- 
mediate effect  on  the  angular  motion  is  nil.  It  is  not  till  it  Las  acted 
long  enough  to  produce  a  perceptible  change  in  the  distance  of  the  dis- 
turbed planet  from  the  sun  that  the  angular  velocity  begins  to  be  sensibly 
affected,  and  it  is  not  till  its  whole  outward  action  has  been  exerted  (i.  e. 
over  the  whole  interval  from  neutral  point  to  neutral  point)  that  its  maxi- 
mum effect  in  lifting  the  disturbed  planet  away  from  the  sun  has  been 
produced,  and  the  full  amount  of  diminution  in  angular  velocity  it  is 
capable  of  causing  has  been  developed.  This  continues  to  act  in  pro- 
ducing a  retardation  in  longitude  long  after  the  normal  force  itself  lias 
reversed  its  action,  and  from  a  powerful  outward  force  has  become  a  feeble 
inward  motion.  A  certain  portion  of  this  perturbation  is  incident  on  tbi 
epoch  in  the  mode  described  in  art.  (731.)  et  seq.,  and  permanently  dis- 
turbs the  mean  motion  from  what  it  would  have  been,  had  Neptune  no 
existence.  The  rest  of  its  effect  is  compensated  in  a  single  synodic 
revolution,  not  by  the  reversal  of  the  action  of  the  force  (for  that  reversed 
action  is  far  too  feeble  for  this  purpose),  but  by  the  effect  of  the  perma- 
nent alteration  produced  in  the  excentricity,  which  (the  axis  being  un- 
changed) compensates  by  increased  proximity  in  one  part  of  the  revolu- 
tion, for  increased  distance  in  the  other.  Sufficit  ■  t  time  has  not  yet 
elapded  since  the  conjunction  to  bring  out  into  full  evidence  the  influonoe 
of  this  force.  Still  its  commencement  is  quite  unequivocally  marked  in 
the  more  rapid  descent  of  our  curve  Ji(]f.  4,  subsequent  to  the  conjunction 
than  ascent  previous  to  that  epoch,  which  indicates  the  commencement  of 
a  series  of  undulations  in  its  future  course  of  an  elliptic  character,  con- 
sequent on  the  altered  excentricity  and  perihelion  (the  total  and  ultimate 
effect  of  this  constituent  of  the  disturbing  force)  which  will  be  maintained 
till  within  about  20  years  from  tho  next  conjunction,  with  the  exception, 
perhaps,  of  some  trifling  inequalities  about  the  time  of  the  opposition, 


PERTURBATIONS  OF  URANUS  BY  NEPTUNE. 


487 


similar  in  character,  but  far  inferior  in  magnitude  to  those  now  undei 
discussion. 

(776.)  Posterity  will  hardly  credit  that,  with  a  full  knowledge  of  all 
the  circumstances  attending  this  great  discovery  —  of  the  calculations  of 
Lcvcrrier  and  Adams  —  of  the  communication  of  its  predicted  place  to 
Dr.  Gallo  — and  of  the  new  planet  being  actuully  found  by  him  in  that 
place,  in  the  remarkable  manner  above  commemorated;  not  only  have 
doubts  been  expres-ied  as  to  the  validity  of  the  calculations  of  those 
geometers,  and  the  legitimacy  of  their  conclusions,  but  these  doubts  have 
been  carried  so  far  as  tu  lead  the  objectors  to  attribute  the  acknowledged 
fact  of  a  planet  previously  unknown  occupying  that  precise  place  in  the 
heavens  at  that  precise  time,  to  sheer  accident  1'  What  share  accident 
may  have  had  in  the  successful  issue  of  the  calci'.lations,  we  presume  the 
reader,  after  what  has  been  said,  will  have  little  difficulty  in  satisfying 
biiusclf.  As  regards  the  time  when  the  discovery  was  made,  much  has 
also  been  attributed  to  fortunate  coincidence.  The  following  considera- 
tions will,  we  apprehend,  completely  dissipate  this  idea,  if  still  lingering 
in  the  mind  of  any  one  at  all  conversant  with  the  subject.  The  period 
of  Uranus  being  84-0140  years,  and  that  of  Neptune  164-6181,  their 
synodic  revolution  (art.  418.),  or  the  interval  between  two  successive  con- 
junctions, is  171-58  years.  The  late  conjunction  having  taken  place 
about  the  beginning  of  1822  j  that  next  preceding  must  have  happmed 
in  1649,  or  more  than  40  years  before  the  first  recorded  observation  of 

'  These  doubts  soem  to  have  originated  partly  in  the  great  disagreement  between  the 
predicted  and  real  elements  of  Neptune,  partly  in  the  near  {ponsibly  precise)  commen- 
Burability  of  the  mean  motions  of  Neptune  and  Uranus.  We  conceive  them  however 
to  be  founded  in  a  total  misconception  of  the  nature  of  the  problem,  which  was  not, 
from  such  obviously  uncertain  indications  as  the  observed  discordances  could  give,  to 
determine  as  astronomical  quantities  the  axis,  exccntricity  and  mass  of  the  disturbing 
planet;  but  practically  to  discover  where  to  look  fur  it:  when,  if  once  found,  these 
elements  would  be  far  better  ascertained.  To  do  this,  any  atis,  excentncity,  perihu- 
lion,  and  maxs,  however  wide  of  the  truth,  which  would  represent,  even  roughly  the 
amount,  but  with  tolerable  correctness  the  direction  of  the  disturbing  force  during  the 
very  moderate  interval  when  the  departures  from  theory  were  really  considerable, 
would  equally  serve  their  purposes ;  and  with  an  excent.  icity,  mass,  and  p'  rihelion  die 
posablo,  it  is  obvious  that  any  assumption  of  the  axis  between  the  liml.j  30  and  J8, 
nay,  even  with  n  much  wider  inferior  limit,  would  serve  the  pur|iose.  In  his  attempt 
to  assign  an  inferior  limit  to  the  axis,  and  in  the  value  so  assigned,  M.  Leverrier,  it 
must  be  admitted,  was  not  successful.  Mr.  Adams,  on  the  other  hand,  influenced  Ly 
no  considerations  of  the  kind  which  appear  to  have  weighed  with  his  brother  geometer, 
fixed  ultimately  (as  we  have  seen)  on  an  axis  not  very  egregiously  wrong.  Still  it  were 
to  be  wished,  for  the  satisfaction  of  all  parties,  that  some  one  would  undertake  the 
problem  de  novo,  employing  formulse  not  liable  to  the  passage  through  infinity,  which, 
technically  speaking,  hampers,  or  may  be  supposed  to  hamper  the  continuous  appiica- 
lion  of  the  usual  perturbational  formulse  when  cases  of  commensurability  occur. 


'i 


1' 


in 

11 

II 

Pi 

1 1 


4S8 


OUTLINES   OP  ASTRONOMY. 


Uranus  in  1690,  to  eay  nothing  of  its  discovery  as  a  planot.  In  1000, 
then,  it  must  have  been  effectually  out  of  reach  of  any  porturbativo  iiiflu- 
ence  worth  considering,  and  so  it  remained  during  the  whole  interval  from 
thence  to  1800.  From  that  time  the  effect  of  perturbation  began  to  bo- 
come  sensible,  about  1805  prominent,  and  in  1820  had  nearly  reuchod  its 
maximum.  At  this  epoch  an  alarm  was  sounded.  The  maximum  was 
not  attained, — the  event,  so  important  to  astronomy,  was  still  in  progress 
of  devolopement,  —  when  the  fact  (any  thing  rather  than  a  striking  one) 
was  noticed,  and  made  matter  of  complaint.  But  the  time  for  discuHsing 
its  cause  with  any  prospect  of  success  was  not  yot  come.  Every  thing 
turns  upon  the  precise  determination  of  the  epoch  of  the  maximum, 
when  the  perturbing  and  perturbed  planet  wore  in  conjunction,  and  upon 
the  law  of  increase  and  diminution  uf  the  perturbation  itself  on  cither 
side  of  that  point.  Now  it  is  always  difficult  to  assign  the  time  of  the 
occurrence  of  a  maximum  by  observations  liable  to  errors  bearing  a  ratiu 
fur  from  inconsiderable  to  the  whulo  quantity  observed.  Until  the  lapse 
of  some  years  from  1822  it  would  have  been  impossible  to  have  fixed  that 
epoch  with  any  certainty,  and  as  respects  the  law  of  degradation  and  totul 
arc  of  longitude  over  which  the  sensible  perturbations  extend,  wo  are 
hardly  yet  arrived  at  a  period  when  this  can  be  said  to  bo  completely  de- 
terminable from  observation  alone.  In  all  this  we  see  nothing  of  acci- 
dent, unless  it  be  accidental  that  an  event  which  must  have  happened 
between  1781  and  1953,  actually  happened  in  1822;  and  that  we  live  in 
an  age  when  astronomy  has  reached  that  perfection,  and  its  cultivators 
exercise  that  vigilance  which  neither  permit  such  an  event,  nor  its  scientific 
importance,  to  pass  unnoticed.  The  blossom  had  been  watched  with  in- 
terest in  its  developement,  and  the  fruit  was  gathered  in  the  very  moment 
of  maturity.' 

•  The  student  who  may  wish  to  see  the  perturbations  of  Uranus  produced  by  Nep- 
tune, as  computed  from  a  knowledge  of  'he  elements  and  mass  of  that  planet,  such 
as  we  now  know  to  be  pretty  near  the  truth,  will  find  them  stated  at  length  from  the 
calculations  of  Mr.  Walker,  (of  Washington,  U.  S.)  in  the  "  Proceedings  of  the  Amer- 
icafi  Academy  of  Arts  and  Sciences,"  vol.  i.  p.  334,  et  seq.  On  examining  the  com- 
parisons of  the  results  of  Mr.  Walker's  formulae  with  those  of  Mr.  Adam's  theory  in 
p.  342,  he  will  perhaps  be  surprised  at  the  enormous  difference  between  the  actions  ot 
Neptune  and  Mr.  Adam's  "  hypothetical  planet"  on  the  longitude  of  Uranus.  This 
is  easily  explained.  Mr.  Adam's  perturbations  are  deviations  from  Bouvard's  orbii  ot 
Uratius,  as  it  stood  immediately  previous  to  the  late  conjunction.  Mr.  Walker's  are 
the  deviations  from  a  mean  or  undisturbed  orbit  freed  from  the  influence  of  the  /ong 
inequality  resulting  from  the  near  commensurability  of  the  motions. 


/ 


OF  SIDEREAL  ASTBONOMY. 


439 


il     '! 


-fl 


PART  HI. 

OF    SIDEREAL    ASTRONOMT. 
CHAPTER  XV. 

OP  THE  FIXED  STARS.  —  THEIR  OliASSIFlOATION  BY  MAGNITUDES. — 
PHOTOMETRIC  SCALE  OP  MAGNITUDES. — CONVrNTIONAL  OR  VULGAR 
SCALE. —  PHOTOMETRIC  COMPARISON  OP  STA;  S. —  DISTIIBUTION  OP 
STARS  OVER  THE  HEAVENS. — OP  THE  MILKY  WAY  t  '  GALAXY. — 
ITS  SUPPOSED  FORM  THAT  OP  A  PLAT  STRATUM  AITIALLY  SUB- 
DIVIDED.— ITS  VISIBLE   COURSE  AMONG   TH^  nONSTELLATIONF ITS 

INTERNAL  STRUCTURE. —  ITS  APPARENTLY  iM  EPINITE  EXTii^T  IN 
CERTAIN  DIRECTIONS. — OP  THE  DISTANCE  OP  THE  FIXED  STARS. — 
THEIR  ANNUAL  PARALLAX.  —  PARALLACTIC  UNIT  OP  SIDEREAL 
DISTANCE. — EFFECT  OF  PARALLAX  ANALOGOUS  TO  THAT  OF  ABER- 
RATION.— HOW  DISTINGUISHED  FROM  IT.  —  DETECTION  OF  PARAL- 
LAX BY  MERIDIONAL  OBSERVATIONS. — HI -VDERSON's  APPLICATION 
TO  a  CENTAURI. — BY  DIFFERENTIAL  OBSERVATIONS. — DISCOVERIES 
OF  BESSEL  AND  8TRUVE.  —  LIST  OF  STARS  IN  WHICH  PARALLAX 
HAS  BEEN  DETECTED. — OP  THE  REAL  MAGNITUDES  OF  THE  STARS. — 
COMPARISON   OP  THEIR  LIGHTS   WITH   THAT   OF  THE   SUN. 

(777.)  Besides  the  bodies  w-  ^  lee  described  in  the  foregoing  chap- 
ters, the  heavens  present  us  with  an  innumerable  multitude  of  other 
objects,  which  are  called  generally  by  the  name  of  stars.  Though  com- 
prehending individuals  differing  from  each  other,  not  merely  in  brightness, 
but  in  many  other  estieniai  points,  they  all  agree  in  one  attribute, — a 
high  degree  of  permanence  as  to  apparent  relative  situation.  This  has 
procured  them  the  title  of  "  fixed  stars ;"  an  expression  which  is  to  be 
understood  in  a  comparative  and  not  an  absolute  sense,  it  being  certain 
that  many,  and  probable  that  all,  are  in  a  state  of  motion,  although  too 
slow  to  be  perceptible  unless  by  means  of  very  delicate  observations,  con- 
tinued during  a  long  series  of  years. 

(778.)  Astronomers  are  in  the  habit  cf  distinguishing  the  stars  into 


'  '1-' 


'1 


11 


Mi 


440 


OUTLINES   OF  ASTRONOMY. 


classes^  according  to  their  apparent  brightness.  These  are  termed  magni- 
tudes. The  brightest  stars  are  said  to  be  of  the  first  magnitude ;  those 
which  fall  so  far  short  of  the.  first  degree  of  brightness  as  to  make  a 
strongly  marked  distinction  are  classed  in  the  second ;  and  so  on  down  to 
the  sixth  or  seventh,  which  comprise  the  smallest  stars  visible  to  the 
naked  eye,  in  the  clearest  and  darkest  night.  Beyond  these,  however, 
telescopes  continue  the  range  of  visibilitj-,  and  magnitndes  from  the  8th 
down  to  the  16th  are  familiar  to  those  who  are  in  the  practice  of  using 
powerful  instruments ;  nor  does  there  seem  the  least  reason  to  assign  a 
limit  to  this  progression ;  every  increase  in  the  dimensions  and  power  of 
instruments,  which  successive  improvements  in  optical  science  have 
attained,  having  brought  into  view  multitudes  innumerable  of  objects 
invisible  before ;  so  that,  for  any  thing  experience  has  hitherto  taught  us, 
the  number  of  the  stars  may  be  really  infinite,  in  the  only  sense  in  which 
we  can  assign  a  meaning  to  the  word. 

(779.)  This  classification  into  magnitudes,  however,  it  must  be  ob- 
served, is  entirely  arbitrary.  Of  a  multitude  of  bright  objects,  difiering 
probably,  intrinsically,  both  in  size  and  in  splendour,  and  arranged  at 
unequal  distances  from  us,  one  must  of  necessity  appear  the  brightest,  one 
next  below  it,  and  so  on.  An  order  of  succession  (relative,  of  course,  to 
our  local  situation  among  them)  must  exist,  and  it  is  a  matter  of  absolute 
indiiference,  where,  in  that  infinite  progression  downwards,  from  the  one 
brightest  to  the  invisible,  we  choose  to  draw  our  lines  of  demarcation.  All 
this  is  a  matter  of  pure  convention.  Usage,  however,  has  established 
such  a  convention  j  and  though  it  is  impossible  to  determine  exactly,  or 
d  priori,  where  one  magnitude  ends  and  the  next  begins,  and  although 
different  observers  have  difiered  in  their  magnitudes,  yet,  on  the  whole, 
astronomers  have  restricted  their  first  magnitude  to  about  23  or  24  prin- 
cipal stars ;  their  second  to  50  or  60  next  inferior ;  their  third  to  about 
200  yet  smaller,  and  so  on ;  the  numbers  increasing  very  rapidly  as  we 
descend  in  the  scale  of  brightness,  the  whole  number  of  stars  already  regis- 
tered down  to  the  seventh  magnitude,  inclusive,  amounting  to  from  12000 
tu  15000. 

(780.)  As  we  do  not  see  the  actual  disc  of  a  star,  but  judge  only  of  its 
brightness  by  the  total  impression  made  upon  the  eye,  the  apparent  "mag- 
nitude" of  any  star  will,  it  is  evident,  depend,  1st,  on  the  star's  distance 
from  us ;  2d,  on  the  absolute  magnitude  of  its  illuminated  surface ;  3d, 
on  the  intrinsic  brightness  of  that  surface.  Now,  as  we  know  nothing  or 
next  to  nothing,  of  any  of  these  data,  and  have  every  reason  for  believing 
that  each  of  them  may  difier  in  different  individuals,  in  the  proportion  of 
many  millions  to  one,  it  is  clear  that  we  are  not  to  expect  much  satisfeo- 


MAGNITUDES   OF  THE   STABS. 


441 


tion  in  any  conclusions  we  may  draw  from  numerical  statements  of  the 
number  of  individuals,  which  have  been  arranged  in  our  artificial  classes 
antecedent  to  any  general  or  definite  principle  of  arrangement.     In  fact, 
astronomers  have  not  yet  agreed  upon  any  principle  by  which  the  magni- 
tudes may  be  photometrically  classed  d  priori,  whether  for  example  a 
scale  of  brightnesses  decreasing  in   geometrical  progression  should  be 
adopted,  each  term  being  one-half  of  the  preceding,  or  one-third,  or  any 
other  ratio,  or  whether  it  would  not  be  preferable  to  adopt  a  scale  decreas- 
iug  as  the  squares  of  the  terms  of  an  harmonic  progression,  i.  e.  according 
to  the  series  1,  \,  \,  Jg,  -^j,  &c.     The  former  would  be  a  purely  photo- 
metric scale,  and  would  have  the  apparent  advantage,  that  the  light  of  a 
star  of  any  magnitude  would  bear  a  fixed  proportion  to  that  of  the  mag- 
nitude next  above  it,  an  advantage,  however,  merely  apparent,  as  it  is  cer- 
tain, from  many  optical  facts,  that  the  unaided  eye  forms  very  difierent 
judgments  of  the  proportions  existing  between  bright  lights,  and  those 
between  feeble  ones.     The  latter  scale  involves  a  physical  idea,  that  of 
supposing  the  scale  of  magnitudes  to  correspond  to  the  appearance  of  a 
first  magnitude  standard  star,  removed  successively  to  twice,  three  times,  Ace., 
its  original  or  standard  distance.     Such  a  scale,  which  would  make  the 
nominal  magnitude  a  sort  of  index  to  the  presumable  or  average  distance, 
on  the  supposition  of  an  equality  among  the  real  lights  of  the  stars, 
would  facilitate  the  expression  of  speculative  ideas  on  the  constitution  of 
the  sidereal  heavens.     On  the  other  hand,  it  would  at  first  sight  appear 
to  make  too  small  a  difference  between  the  lights  in  the  lower  magnitudes. 
For  example,  on  this  principle  of  nomenclature,  the  light  of  a  star  of  the 
seventh  magnitude  would  be  thirty-six  49th3  of  that  of  one  of  the  sixth, 
and  of  the  tenth  81  hundredths  of  the  ninth,  while  between  the  first  and 
the  second  the  proportion  would  be  that  of  four  to  one.    So  far,  however, 
from  this  being  really  objectionable,  it  falls  in  well  with  the  general  tenor 
of  the  optical  facts  already  alluded  to,  inasmuch  as  the  eye  (in  the  ab- 
sence of  disturbing  causes)  does  actually  discriminate  with  greater  preci- 
sion between  the  relative  intensities  of  feeble  lights  than  of  bright  ones, 
so  that  the  fraction  ||,  for  instance,  expresses  quite  as  great  a  step  down- 
wards (physiologically  speaking)  from  the  sixth  magnitude,  as  \  does  from 
the  first.   As  the  choice,  therefore,  so  far  as  we  can  see,  lies  between  these 
two  scales,  in  drawing  the  lines  of  demarcation  between  what  may  be 
termed  the photomctrical  magnitudes  of  the  stars,  we  have  no  hesitation 
in  adopting,  and  recommending  others  to  adopt,  the  latter  system  in  pre- 
ference to  the  former. 

(781.)  The  conventional  magnitudes  actually  in  use  among  astrono- 
mers, so  far  as  their  usage  is  consistent  with  itself,  conforms  moreover 


1'   ! 


iif] 


m 


ium^ 


rj 


442 


OUTLINES  OF  ASTRONOMY. 


very  much  more  nearly  to  this  than  to  the  geometrical  progression.  It  has 
been  shown/  by  direct  photometjrio  measurement  of  the  light  of  a  consi- 
derable number  of  stars,  from  the  first  to  the  fourth  magnitude,  that  if  M 
be  the  number  expressing  the  magnitude  of  a  star  on  the  above  system, 
and  m  the  number  expressing  the  magnitude  of  the  same  star  in  the  loose 
and  irregular  language  at  present  conventionally  or  rather  provisionally 
adopted,  so  far  as  it  can  be  collected  from  the  conflicting  authorities  of 
different  observers,  the  difference  between  these  numbers,  or  M  —  m,  is 
the  same  in  all  the  higher  parts  of  the  scale,  and  is  less  than  half  a  mag- 
nitude (O"'  414).  The  standard  star  assumed  as  the  unit  of  magnitude 
in  the  measurements  referred  to,  is  the  bright  southern  star  a  Centuuri,  a 
star  somewhat  superior  to  Arcturus  in  lustre.  If  we  take  the  distance  of 
this  s^ar  for  unity,  it  follows  that  when  removed  to  the  distances  1414, 
2'414,  3-414,  «&,c.,  its  apparent  lustre  would  equal  those  of  average  stars 
of  the  1st,  2d,  3d,  &c.  magnitudes,  a8  ordinarily  reckoned,  respectively. 
(782.)  The  difference  of  lustre  between  stars  of  two  consecutive  mag- 
nitudes is  so  considerable  as  to  allow  of  many  'ntermediate  gradations 
being  perfectly  well  distinguished.  Hardly  any  two  stars  of  the  first,  or 
of  the  second  magnitude,  would  be  judged  by  an  eye  practised  in  such 
comparisons  to  be  exactly  equal  in  brightness.  Hence,  the  necessity,  if 
anything  like  accuracy  be  aimed  at,  of  subdividing  the  magnitudes  and 
admitting  fractious  into  our  nomenclature  of  brightness.  When  this  ne- 
cessity first  began  to  be  felt,  a  simple  bisection  of  the  interval  was  recog- 
nized, and  the  intermediate  degree  of  brightness  was  thus  designated,  viz. 
1.2  m,  2.3  m,  and  so  on.  At  present  it  is  not  unfrequent  to  find  the  in- 
terval trisected  thus:  1  m,  1.2  m,  2.1  m,  2  m,  &c.  where  the  expression 
1.2  m  denotes  a  magnitude  intermediate  between  the  first  and  second,  but 
nearer  1  than  2;  while  2.1  m  designates  a  magnitude  also  intermediate, 
but  nearer  2  than  1.  This  may  suffice  for  common  parlance,  but  as  this 
department  of  astronomy  progresses  towards  exactness,  a  decimal  subdi- 
vision will  of  necessity  supersede  these  rude  forms  of  expression,  and  the 
magnitude  will  be  expressed  by  an  integer  number,  followed  by  a  decimal 
fraction;  as,  for  instance,  2.51,  which  expresses  the  magnitude  of  y  Gomi- 
norum  on  the  vulgar  or  conventional  scale  of  magnitudes,  by  which  we  at 
once  perceive  that  its  place  is  almost  exactly  half  way  between  the  2d  and 
8d  average  magnitudes,  and  that  its  light  is  to  that  of  in  average  first 
magnitude  star  in  that  scale  (of  which  a  Ononis  in  its  usual  or  normal 
Btate^  may  be  taken  as  a  typical  specimen)  as  1*:  (2*51)*,  and  to  that  of 

'  See  "  Results  of  observations  made  at  the  Cape  of  Good  Hope,  &c.  &c."  p.  371. 
By  the  Author. 

•  In  the  interval  from  1836  to  1839  this  star  underwent  considerable  and  remarkable 
fluctuations  of  brightness. 


PHOTOUBTRIO  SCALE  OF  MAGNITUDES. 


448 


a  Ccntauri  as  1':  (2-924)',  making  its  place  in  the  photometric  scale  (so 
defined)  2-924.  Lists  of  sturs,  northern  and  southern,  comprehending 
those  of  the  vulgar  first,  second  and  third  magnitudes,  with  their  magni* 
tudes  decimally  expressed  in  both  systems,  will  be  found  at  the  end  of 
this  work.  The  light  of  a  star  of  the  sixth  magnitude  may  be  roughly 
stated  as  about  the  hundredth  part  of  one  of  the  first.  Sirius  would  make 
between  three  and  four  hundred  stars  of  that  magnitude. 

(783.)  The  exact  photomctrical  determiaation  of  the  comparative  in- 
tensities of  light  of  the  stars  is  attended  with  many  and  great  difficulties, 
arising  partly  from  their  differences  of  colour ;  partly  from  the  circum- 
stance that  no  invariable  standard  of  artificial  light  has  yet  been  disco- 
vered ;  partly  from  the  physiological  cause  above  alluded  to,  by  which  the 
eye  is  incapacitated  from  judging  correctly  of  the  proportion  of  two  lights, 
and  can  only  decide  (and  that  with  not  very  great  precision)  as  to  their 
equality  or  inequality  j  and  partly  from  other  physiological  causes.     The 
least  objectionable  mode  hitherto  proposed  would  appear  to  be  the  follow- 
ing.    A  natural  standard  of  comparison  is  in  the  first  instance  selected, 
brighter  than  any  of  the  stars,  so  as  to  allow  of  being  equalized  with  any 
of  them  by  a  reduction  of  its  light  optically  effected,  and  at  the  same  time 
either  invariable,  or  at  least  only  so  variable  that  its  changes  can  be  ex- 
actly calculated  and  reduced  to  numerical  estimation.     Such  a  standard  is 
offered  by  the  planet  Jupiter,  which,  being  much  brighter  than  any  star, 
subject  to  no  phases,  and  variable  in  light  only  by  the  variation  of  its  dis- 
tance from  the  sun,  and  which,  moreover,  comes  in  succession  above  the 
horizon  at  a  convenient  altitude,  simultaneously  with  all  the  fixed  stars, 
and,  in  the  absence  of  the  moon,  twilight,  and  other  disturbmg  causes 
(which  fatally  affest  all  observations  of  this  nature),  combmes  all  the  re- 
quisite conditions.     Let  us  suppose,  now,  that  Jupiter  being  at  A  and  the 
star  to  be  compared  with  it  at  B,  a  glass  prism,  C,  is  so  placed  that  the 
light  of  the  planet  deflected  by  total  internal  rejlexion  at  its  base,  shall 
emerge  parallel  to  B  E,  the  direction  of  the  star's  visual  ray.     After  re- 
flexion let  it  be  received  on  a  lens,  D,  in  whose  focus,  F,  it  will  form  a 
small,  bright,  star-like  image,  capable  of  being  viewed  by  an  eye  placed 
at  E,  so  far  out  of  the  axis  of  the  cone  of  diverging  rays  as  to  admit  of 
seeing  at  the  same  time,  and  with  the  same  eye,  and  so  comparing  this 
image  with  the  star  seen  directly.     By  bringing  the  eye  neai'er  to  or 
further  from  the  focus,  F,  the  apparent  brightness  of  the  focal  point  will 
be  varied  in  the  inverse  ratio  of  the  square  of  the  distance,  E  F,  and 
therefore  may  be  equalized^  as  well  as  the  eye  can  judge  of  such  equali- 
ties, with  the  star.    If  this  be  done  for  two  stars  several  times  alternately, 
and  a  mean  of  the  results  taken,  by  measuring  E  F,  their  relative  bright- 


•'' .  l- 


;i'  »-! 


''5. 1 


'IP 


l!.i 


444 


OUTLINES  OP  ASTRONOMY. 


ness  will  be  obtained :  that  of  Jupiter,  the  temporary  standaru  of  com- 
parison, being  altogether  eliminated  from  the  result. 

(784.)  A  moderate  number  of  well  selected  stars  being  thus  photometri- 
cally determined  by  repeated  and  careful  measurements,  so  as  to  afford  an 
ascertained  and  graduated  scale  of  brightness  among  the  stars  themselves, 
the  intermediate  steps  or  grades  of  magnitude  may  be  iSUed  up,  by  insert- 
ing between  them,  according  to  the  judgment  of  the  eye,  other  stars, 
forming  an  ascending  or  descending  sequence,  each  member  of  such  a 
sequence  being  brighter  than  that  below,  and  less  bright  than  that 
above  it;  and  thus  at  length,  a  scale  of  numerical  magnitudes  will 
become  established,  complete  in  all  its  members,  from  Sirius,  the  brightest 
of  the  stars,  down  to  the  least  visible  magnitude.'  It  were  much  to  be 
wished  that  this  branch  of  astronomy,  which  at  present  can  hardly  be  said 
to  be  advanced  beyond  its  infancy,  were  perseveringly  and  systematically 
cultivated.  It  is  by  no  means  a  subject  of  mere  barren  curiosity,  as  will 
abundantly  appear  when  we  come  to  speak  of  the  phaenomena  of  variable 
stars,  and  being  moreover,  one  in  which  amateurs  of  the  science  may 
easily  chalk  out  for  themselves  a  useful  and  available  path,  may  naturally 
be  expected  to  receive  large  and  interesting  accessions  at  their  hands. 

(785.)  If  the  comparison  of  the  apparent  magnitudes  of  the  stars  with 
their  numbers  leads  to  no  immediately  obvious  conclusion,  it  is  otherwise 
when  we  view  them  in  connexion  with  their  local  distribution  over  the 
heavens.  If  indeed  we  confine  ourselves  to  the  three  or  four  brightest 
classes,  we  shall  find  them  distributed  with  a  considerable  approach  to 

'  For  the  method  of  combining  and  treating  such  sequences,  where  accumulated 
in  considerable  numbers,  so  as  to  eliminate  from  their  results  the  influence  of  erroneous 
judgment,  atmospheric  circumstancos,  &c.,  which  often  give  rise  to  contradictory 
arrangements  in  the  order  of  stars  differing  but  little  in  magnitude,  as  well  as  for  an 
account  of  a  series  of  photometric  comparisons  (in  which,  however,  not  Jupiter,  but 
the  moon  was  used  as  an  intermediate  standard),  see  the  work  above  cited,  note  on  p. 
353     (Results  of  Observations,  &,c.) 


Kf, '.,:-.  I.- ■ 


GENERAL  FORM  OF  THE  GALAXY. 


445 


impartiality  over  the  sphere :  a  marlced  preference  however  being  observa- 
ble, especially  in  the  southern  hemisphere,  to  a  zone  or  belt,  following  the 
direction  of  a  great  circle  passing  through  <  Orionis  and  a  Crucis.  But 
if  we  take  in  the  whole  amount  visible  to  the  naked  eye,  we  shall  perceive 
a  great  increase  of  number  as  we  approach  the  borders  of  the  Milky  Way. 
And  when  we  come  to  telescopic  magnitudes,  we  find  them  crowded 
beyond  imagination,  along  the  extent  of  that  circle,  and  of  the  branches 
which  it  sends  off  from  it ;  so  that  in  fact  its  whole  light  is  composed  of 
nothing  but  stars  of  every  magnitude,  from  such  as  are  visible  to  the 
naked  eye  down  to  the  smallest  point  of  light  perceptible  with  the  best 
telescopes. 

(786.)  These  phaenomena  agree  with  the  supposition  that  the  stars  of 
our  firmament,  instead  of  being  scattered  in  all  directions  indifferently 
through  space,  form  a  stratum  of  which  the  thickness  is  small,  in  com- 
parison with  its  length  and  breadth  j  and  in  which  the  earth  occupies  a 
place  somewhere  about  the  middle  of  its  thickness,  and  near  the  point 
where  it  subdivides  into  two  principal  laminae,  inclined  at  a  small  angle  to 
each  other  (art.  302).  For  it  is  certain  that,  to  an  eye  so  situated,  the 
apparent  density  of  the  stars,  supposing  them  pretty  equally  scattered 
through  the  space  they  occupy,  would  bo  least  in  a  direction  of  the  visual 
ray  (as  S  A),  perpendicuhr  to  the  lamina,  and  greatest  in  that  jf  its 
breadth,  as  S  B,  S  C,  S  D ;  increasing  rapidly  in  passing  from  one  to  the 
other  direction,  just  as  we  see  a  slight  haze  in  the  atmosphere  thickening 
into  a  decided  fog-bank  near  the  horizon,  by  the  rapid  increape  of  the 
mere  length  of  the  visual  ray.  Such  is  the  view  of  the  construction  of 
the  starry  firmament  taken  by  Sir  William  Herscbei,  whose  powerful 


telescopes  first  effected  a  complete  analysis  of  this  wonderful  zone,  and 
demonstrated  the  fact  of  its  entirely  consisting  of  stars.'  So  crowded  are 
they  in  some  parts  of  it,  that  by  counting  the  stars  in  a  single  field  of  his 

'Thomas  Wright  of  Durham  (Theory  of  the  Universe,  London,  1750)  appeara  bo 
early  as  1734  to  have  entertained  the  same  general  view  as  to  the  constitution  of  the 
Milky  Way  and  starry  firmament,  founded,  quite  in  the  spirit  of  just  astronomical 
speculution,  on  a  partial  resolution  of  a  portion  of  it  with  a  "  one-foot  reflector"  (a 
reflector  one  foot  in  focal  length).  See  an  account  of  this  rare  work  by  M.  do  Morgan 
in  Phil.  Mag.  Ser.  3.  xxxii.  p.  241.  et  seq. 


f'!':i 


*; 


p 


;ki 


11 


m 


1 ) 


446 


OUTLINES  OP  ASTRONOMY. 


telescope,  he  was  led  to  conclude  that  50,000  had  passed  under  his  review 
in  a  zouC  two  degrees  in  breadth,  during  a  single  hour's  observation.  Id 
that  part  of  the  milky  way  which  is  situated  in  lOh  30m  R  A  and  between 
the  147th  and  150th  degree  of  N  P  D,  up'j/anit!  of  6000  stars  have  been 
reckor.ed  to  exist  in  a  square  degroe.  Thu  imaituse  iiatances  at  which 
the  ren'oter  regions  must  be  situateij  wili  »ufficieutly  'iccunt  for  the  vast 
predominance  of  small  magnitudes  v, l)icU  ajv  ob^'^-  ved  );>  i' 

(787.)  The  course  of  tho  Milky  Way  as  ti-uced  th/i!iij;^h  the  heavens 
by  the  unaided  eye,  riegleotin^i::^  occaeicv.!  deviations  and  following  the  line 
of  its  greatest  brigbtuf^^a  as  well  as  its  varying  breadth  and  intensity  will 
permit,  conforms  nearly  to  that  of  a  grca?  circle  inclnied  \t  an  angle  of 
about  03**  to  the  equinoctial,  nnd  cutting  that  circlo  in  R  A  0^  47m  and 
19A  4:7m,  so  iiiat  its  novtiierr.  and  southern  ]>o'^«  respectively  are  situated 
in  R.  A.  Uh  47m  N  P  D  68  '  and  R.  A,  m,  4V.<'  N  P  D  117°.  Through- 
o«t  the  region  where  it  is  so  remarkably  suodivided  (art.  186),  this  great 
<*ii*cle  hciJs  an  intermediate  situation  between  the  two  great  streams  j  with 
u  r.teafer  approximation  however  to  the  brighter  and  continuous  stream, 
than  to  the  fainter  and  interrupted  one.  If  we  trace  its  course  in  order 
of  right  ascension,  we  find  it  traversing  the  constellation  Cassiopeia,  its 
brightest  part  passing  about  two  d-i^rees  to  the  north  of  the  star  6  of  that 
constellation,  i.  e.  in  about  62°  of  north  declination,  or  28°  N  P  D. 
Passing  thence  between  y  and  »  Cassiopeiae  it  sends  off  a  branch  to  the 
south-preceding  side,  towards  »  Persci,  very  conspicuous  as  far  as  that 
star,  prolonged  faintly  towards  b  of  the  same  constellation,  and  possibly 
traceable  towards  the  Hyades  and  Pleiades  as  remote  outliers.  The  main 
stream  however  (which  is  here  very  faint),  passes  on  through  Auriga,  over 
the  three  remarkable  stars, «,  Cj  •?>  of  that  constellation  preceding  Capella, 
called  the  Hoedi,  preceding  Capella,  between  the  feet  of  Gemini  and  the 
horns  of  the  Bull  (where  it  intersects  the  ecliptic  nearly  in  the  Solstitial 
Colure)  and  thence  over  the  club  of  Orion  to  the  neck  of  Monoceros, 
intersecting  the  equinoctial  in  R.  A.  6A  54m.  tip  to  this  point,  from 
the  offset  in  Perseus,  its  light  is  feeble  and  indefinite,  but  thenceforward 
it  receives  a  gradual  accession  of  brightness,  and  where  it  passes  through 
the  shoulder  of  Monoceros  and  over  the  head  of  Canis  Major  it  presents 
abroad,  moderately  bright,  very  uniform,  and  to  the  naked  eye,  starless 
stream  up  to  the  point  where  it  enters  the  prow  of  the  ship  Argo,  nearly 
on  the  southern  tropic'     Here  it  again  subdivides  (about  the  star  m 

*  In  reading  this  description  a  celestial  globe  will  be  a  necessary  companion.  It  may 
lie  thought  needless  to  detail  the  course  of  the  Milky  Way  verbally,  since  it  is  mapped 
down  on  all  celestial  charts  and  globes.  But  in  the  generality  of  them,  indeed  in  all 
whien  have  come  to  our  knowledge,  this  is  done  so  very  loosely  and  incorrectly,  as  by 
no  means  to  dispense  with  a  verbal  description. 


COURSE  OF   THE  VIA  LACTEA  TRACED. 


447 


Puppis),  sending  off  a  narrow  and  winding  branch  on  the  preceding  side 
as  far  as  y  Argfts,  where  it  terminates  abruptly.  The  main  stream  pur- 
sues its  southward  course  to  the  123d  parallel  of  N  P  D,  where  it  diffuses 
itself  broadly  and  again  subdivides,  opening  out  into  a  wide  fan-like  ex- 
panse nearly  20°  in  breadth  formed  of  interlacing  branches,  all  which 
terminate  abruptly,  in  a  line  drawn  nearly  through  x  and  y  Argfls. 

(788.)  At  this  place  the  continuity  of  the  Milky  Way  is  interrupted 
by  a  wide  gap,  and  where  it  recommences  on  the  opposite  side  it  is  by  a 
somewhat  cimilar  fan-shaped  assemblage  of  branches  which  converge  upon 
the  bright  star  fj  Argfis.  Thence  it  crosses  the  hind  feet  of  the  Centaur, 
forming  a  curious  and  sharply  defined  semicircular  concavity  of  small 
radius,  and  enters  the  Cross  by  a  very  bright  neck  or  isthmus  of  not  more 
than  3  or  4  degrees  in  breadth,  being  the  narrowest  portion  of  the  Milky 
Way.  After  this  it  immediately  expands  into  a  broad  p.rd  bright  mass, 
enclosing  the  stars  a  and  |3  Crucis,  and  (3  Centauri,  and  extending  almost 
np  to  a  of  the  latter  constellation.  In  the  midst  of  this  bright  mass,  sur- 
rounded by  it  on  all  sides,  and  occupying  about  half  its  breadth,  occurs  a 
singular  dark  pear-shaped  vacancy,  so  conspicuous  and  remarkable  as  to 
attract  the  notice  of  the  most  superficial  gazer,  and  to  have  acquired  among 
the  early  southern  navigators  the  uncouth  but  expressive  appellation  of 
the  coal-sack.  In  this  vacancy  which  is  about  8°  in  length,  and  5°  broad, 
only  one  very  small  star  visible  to  the  naked  eye  occurs,  though  it  is  far 
from  devoid  of  telescopic  stars,  so  that  its  striking  blackness  is  simply  due 
to  the  effect  of  contrast  with  the  brilliant  ground  with  which  it  is  on  all 
sides  surrounded.  This  is  the  place  of  nearest  approach  of  the  Milky 
Way  to  the  South  Pole.  Throughout  all  this  region  its  brightness  is  very 
striking,  and  when  compared  with  that  of  its  more  northern  course  already 
traced,  conveys  strongly  the  impression  of  greater  proximity,  and  would 
almost  lead  to  a  belief  that  our  situation  as  spectators  is  separated  on  all 
sides  by  a  considerable  interval  from  the  dense  body  of  stars  composing 
the  Galaxy,  which  in  this  view  of  the  subject  would  come  to  be  considered 
as  a  flat  ring  of  immense  and  irregular  breadth  and  thickness,  within 
which  we  are  excentrically  situated,  nearer  to  the  southern  than  to  the 
northern  part  of  its  circuit. 

(789.)  At  a  Centauri,  the  Milky  Way  again  subdivides',  sending  off  a 
great  branch  of  nearly  half  its  breadth,  but  which  thins  off  rapidly,  at  an 
angle  of  about  20°  with  its  general  direction,  towards  the  preceding  side, 
to  •?  and  d  Lupi,  beyond  which  it  loses  itself  in  a  narrow  and  faint  stream- 
let. The  main  stream  passes  on  increasing  in  breadth  to  y  Normse,  where 
it  makes  an  abrupt  elbow  and  again  subdivides  into  one  principal  and  con 

'  All  the  maps  and  globes  place  this  subdivision  at  /3  Centauri,  but  erroneously. 


In;  fiii 


'»'    VA  ■ 


ii' 


448 


OUTLINES   or  ASTRONOMY. 


tinuous  stream  of  very  irregulai:  breadth  and  brightness  on  the  following 
sido,  and  a  complicated  system  of  interlaced  streaks  and  masses  on  the 
preceding,  which  covers  the  tail  of  Scorpio,  and  terminates  in  a  vast  and 
faint  cifusion  over  the  whole  extensive  region  occupied  by  the  preceding 
leg  of  Opliiuchus,  extending  northwards  to  the  parallel  of  103°  N  P  D, 
beyond  which  it  cannot  be  traced;  a  wide  interva  of  14°,  free  from  all 
appearance  of  nebulous  light,  separating  it  from  the  great  branch  on  the 
north  side  of  the  equinoctial  of  which  it  is  usually  represented  as  a  con- 
tinuation. 

(790.)  Eeturning  to  the  point  of  separation  of  this  great  branch  from 
the  main  stream,  let  us  now  pursue  the  course  of  the  latter.  Making  an 
abrupt  bend  to  the  following  side,  it  passes  over  the  stars  t  Aroo,  B  and  * 
Scorpii,  and  y  Tubi  to  y  Sagittarii,  where  it  suddenly  collects  into  a  vivid 
oval  mass  about  6°  in  length  and  4°  in  breadth,  so  excessively  rich  in 
stars  tha.  a  very  moderate  calculation  makes  their  number  exceed  100,000. 
Northward  of  this  mass,  this  stream  crosses  the  ecliptic  in  longitude  about 
276°,  and  proceeding  along  the  bow  of  Sagittarius  into  Antinous  has  its 
course  rippled  by  three  deep  concavities,  separated  from  each  .>ther  by 
remarkable  protuberances,  of  which  the  larger  and  brighter  (situated 
between  Flamstead's  stars  3  and  6  Aquilae)  forms  the  most  conspicuous 
patch  in  the  southern  portion  of  the  Milky  Way  visible  in  our  latitudea. 

(791.)  Crossing  the  equinoctial  at  the  19th  hour  of  right  ascension,  it 
next  runs  in  an  irregular,  patchy,  and  winding  stream  through  Aquila, 
Sagitta  and  Vulpecula  up  to  Cygnus ;  at «  of  which  coustellation  its  con- 
tinuity is  interrupted,  and  a  very  confused  and  irregular  region  commences, 
marked  by  a  broad  dark  vacuity,  not  unlike  the  southern  "  coal-sack," 
occupying  the  space  between  «,  a,  and  y  Cygni,  which  serves  as  a  kind  of 
centre  for  the  divergence  of  three  greut  streams;  one,  which  we  have 
already  traced ;  a  second,  the  continuation  of  the  first  (across  the  interval) 
from  a  northward,  between  Lacerta  and  the  head  of  Cepheus  to  the  point 
in  Cassiopeia  whence  we  set  out,  and  a  third  branching  off  from  y  Cygni, 
very  vivid  and  conspicuous,  running  off  in  a  southern  direction  through  |3 
Cygni,  and  s  Aquilae  almost  to  the  equinoctial,  where  it  loses  itself  in  a 
region  thinly  sprinkled  with  stars,  w^hers  in  some  maps  the  modern  con- 
stellation Taurus  Poniatovii  is  placed.  This  is  the  branch  whi.^h,  if  con- 
tinued across  the  equinoctial,  might  be  supposed  to  unite  with  the  great 
southern  effusion  in  Ophiuchus  already  noticed  (art.  789).  A  considerable 
offset,  or  protuberant  appendage,  is  also  thrown  off  by  the  northern  stream 
from  the  head  of  Cepheus  directly  towards  the  pole,  occupying  the  greater 
part  of  the  quartile  formed  by  o,  /3,  »,  and  h  of  that  ooustellation. 


COURSE   OP  THE  VIA   LACTEA  TRACED. 


449 


(792.)  Wo  have  been  sonicwliat  circumstantial  in  describing  the  course 
and  principal  features  of  the  Via  Lactea,  not  only  because  there  does  not 
occur  anywhere  (so  far  as  wo  know)  any  correct  account  of  it,  but  chiefly 
by  reason  of  its  high  interest  in  sidereal  astronomy,  and  that  the  reader 
may  perceive  how  very  diflicult  it  must  necessarily  be  to  form  any  just 
conception  of  the  real,  sol'     form,  as  it  exists  in  space,  of  an  object  so 
complicated,  and  which  we  see  from  a  point  of  view  so  uafavonrnble. 
The  difficulty  is  of  the  same  kind  which  we  experience  whan  we  set  our- 
selves to  conceive  tha  real  shape  of  an  auroral  arch  or  of  the  clouds,  but 
far  greaior  in  degree,  because  we  know  the  laws  which  regulate  the  forma- 
tion of  the  latter,  and  limit  them  to  certain  conditions  of  altitude  -  -  be- 
cause their  motion  presents  them  to  us  in  various  aspects,  but  chiefly 
because  we  contemplate  them  from  a  station  considerably  below  their 
general  plane,  so  as  to  allow  of  our  mapping  out  some  kind  of  groimd- 
plan  of  their  shape.     All  these  aids  are  wanting  when  we  attempt  to  reap 
and  model  out  the  Galaxy,  and  beyond  the  obvious  conclusion  that  its 
form  must  be,  generally  speaking,  flat^  and  of  a  thickness  small  in  com- 
parison with  its  area  in  length  and  breadth,  the  laws  of  perspective  afford 
us  little  further  assistance  iu  the  inquiry.     Probability  may,  it  is  true, 
here  and  there  enlighten  us  as  to  certain  features.     Thus  when  we  see,  as 
in  the  coal-sack,  a  sharply  defined  oval  space  free  from  stars,  insulated  in 
the  midst  of  a  uniform  band  of  not  much  more  than  twice  its  breadth,  it 
would  seem  much  less  probable  that  a  conical  or  tubular  hollow  traverses 
the  whole  of  a  starry  stratum,  continuously  extended  from  the  eye  out- 
wards, than  that  a  distant  mass  of  comparatively  moderate  thickness 
should  be  simply  perforated  from  side  to  side,  or  that  an  oval  vacuity 
should  be  seen  foreshortened  in  a  distant  foreshortened  area,  not  really 
exceeding  two  or  three  times  its  own  breadth.     Neither  can  we  without 
obvious  improbability  refuse  to  admit  that  the  long  lateral  offsets  which  at 
so  many  places  quit  the  main  stream  and  run  out  to  great  distances,  are 
either  planes  seen  edgeways,  or  the  convexities  of  curved  surfaces  viewed 
tarigentially,  rather  than  cylindrical  or  columnar  excrescences  bristling  up 
obliquely  from  the  general  level.     And  in  the  same  spirit  of  probable 
Burmise  we  may  account  for  the  intricate  reticulations  above  described  as 
existing  in  the  region  of  Scorpio,  rather  by  the  accidental  crossing  of 
streaks  thus  originating,  at  very  different  distances,  or  by  a  cellular  struc- 
ture of  the  mass,  than  by  real  intersections.     Those  cirrous  clouds  which 
are  often  seen  in  windy  weather,  convey  no  unapt  impression  either  of  the 
kind  of  appearance  in  question,  or  of  the  structure  it  suggests.     It  is  to 
other  indications  howe  :er,  and  chiefly  to  the  telescopic  examination  of  its 
intimate  constitution,  and  to  the  law  of  the  distribution  of  stars,  not  only 
29 


m 


m."^ 


\i\ 


if 


m 


m 


din 


11 


\  I 


'X  '' ' 


450 


OUTLINES   OP  ASTRONOMY. 


within  its  bosom,  but  geucrully  over  the  heavens,  that  we  must  look  for 
more  definite  knowledge  respecting  its  true  form  and  extent. 

(798.)  It  is  on  obscrvatious  of  this  latter  class,  and  not  on  merely 
speculative  or  conjectural  views,  that  the  generalization  in  Art.  780,  whicli 
refers  tha  phajuomena  of  the  starry  firmament  to  the  system  of  the  Ga- 
lasy  as  their  embodying  fact,  is  brought  to  depend,  The  process  of 
"  gcuging  "  the  heavens  was  devised  by  Sir  W.  Horschel  for  this  purpose. 
It  coisisted  in  simply  counting  the  stars  of  all  magnitudes  which  occur  in 
single  fields  of  view,  of  15'  in  diameter,  visible  through  a  reflecting  tele- 
scope of  18  inches  aperture,  and  20  feet  focal  length,  with  a  magnifying 
power  of  180° :  the  points  of  observation  being  very  numerous  and  taken 
indiscriminately  in  every  part  of  the  surface  of  the  sphere  visible  in  our 
latitudes.  On  a  comparison  of  many  hundred  such  "gauges"  or  local 
enumerations  it  appears  that  the  density  of  star-light  (or  the  number  of 
stars  existing  on  an  average  of  several  such  enumerations  in  any  one  im- 
mediate neighbourhood)  is  least  in  the  polo  of  the  Galactic  circle,^  and 
increases  on  all  sides,  with  the  Galactic  polar  distance  (and  that  nearly 
equally  in  all  directions)  down  to  the  Milky  Way  itself,  where  it  attains 
its  maximum.  The  progressive  rate  of  increase  in  proceeding  from  the 
pole  is  at  first  slow,  but  becomes  more  and  more  rapid  as  we  approach  the 
plane  of  that  circle  according  to  a  law  of  which  the  following  numbers, 
deduced  by  M.  Struvo  from  a  careful  analysis  of  all  the  gauges  in  qaes- 
tion,  will  afford  a  correct  idea. 


QalacUc^  North  Polar  UbitaiiM. 

0« 
15^ 
30° 
45° 
C0° 
75° 
90° 


Av«ra)ie  Number  of  dtara  in  a 
I'iold  16'  In  Diameter. 

415 

4-68 

6  52 

10-36 

17-68 

30-30 

122-00 


From  which  it  appears  that  the  mean  density  of  the  stars  in  the  galactic 
circle  exceeds  in  a  ratio  of  very  nearly  30  to  1  that  in  its  pole,  and  in  a 
proportion  of  more  than  4  to  1  that  in  a  direction  15°  inclined  to  its 
plane. 

'  From  y«Xo,  >i!X««roj,  milk ;  meaning  the  great  circle  spoken  ol'  in  Art.  787,  to 
which  the  course  of  the  Via  Lactea  most  nearly  conforms.  Every  subject  has  its  tech- 
nical or  conventional  terms,  by  whose  use  circumlocution  is  avoided,  and  ideas  ren- 
deied  definite.  This  circle  is  to  sidereal  what  the  invariable  ecliptic  is  to  planetary 
Mtrnnomy  —  a  plane  of  ultimate  reference,  the  ground-plane  of  the  sidereal  system. 

'  Etudes  d' Astronomic  Stellaire,  p,  71. 


LAW 


DISTRIBUTION   OP  THE   STARS. 


451 


(794.)  Theao  numbers  fully  bear  out  the  statement  in  Art.  786,  and 
even  draw  closer  the  resemblance  by  which  that  statement  is  there  illus* 
trated.  For  the  rapidly  increasing  density  of  a  fog-bank  as  the  visual  ray 
is  dopr«sscd  towards  the  plane  of  the  horizon  ia  a  consequence  not  only  of 
the  mere  increase  in  length  of  the  foggy  space  traversed,  but  also  of  an 
actual  increase  of  density  in  the  fog  itself  in  its  lower  strata.  Now  this 
very  conclusion  follows  from  a  comparison  inter  ae  of  the  numbers  above 
oet  down,  as  jM.  Struve  has  clearly  shown  from  a  mathematical  analysis 
of  the  empirical  formula,  which  faithfully  represents  their  law  of  progres- 
Bion,  and  of  which  he  states  the  result  in  the  following  table,  expressing 
the  densities  of  the  stars  at  the  respective  distances,  1,  2,  8,  &c.,  from  the 
galactic  plane,  taking  the  mean  densit;^  ^f  the  stars  in  that  plane  itself 
for  unity. 


Dintanceii  from  the 
Qalnctlo  Plane. 

Denalty  of  Staru. 

Digtanoea  from  the 
Qalactio  Plane. 

Density  of  Stan. 

0-00 
0-06 
0-10 
0-20 
0-30 
0-40 

1-00000 
0-48568 
0-33288 
0-23895 
0-17980 
0-13021 

0-50 
0-60 
0-70 
0-80 
0-866 

008646 
0-05510 
0-03079 
0-01414 
0-00532 

The  unit  of  distance,  of  which  the  first  column  of  this  table  expresses 
fractional  parts,  is  the  distance  at  which  such  a  telescope  is  capable  of 
rendering  just  visible  a  star  of  average  magnitude,  or,  as  it  is  termed,  its 
space-penetrating  power.  As  we  ascend  therefore  from  the  galactic  plane 
into  this  kind  of  stellar  atmosphere,  we  perceive  that  the  density  of  its 
parallel  strata  decreases  with  great  rapidity.  At  an  altitude  above  that 
plane  equal  to  only  one-twentieth  of  the  telescopic  limit,  it  has  already 
diminished  to  one-half,  and  at  an  altitude  of  0*866,  to  hardly  more  than 
one-two-hundredth  of  its  amount  in  that  plane.  So  far  as  we  can  perceive 
there  is  no  flaw  in  this  reasoning,  if  only  it  be  granted,  1st,  that  the  level 
planes  are  continuous,  and  of  equal  density  throughout;  and,  2dly,  that 
an  absolute  and  definite  limit  is  set  to  tele^^copk  lision,  beyond  which,  if 
stars  exist,  they  elude  our  sight,  and  are  io  ut.  as  if  they  existed  not :  a 
postulate  whose  probability  the  reader  will  be  in  a  better  condition  to  esti- 
mate, when  in  possession  of  some  other  particulars  respecting  the  consti. 
tution  of  the  Galaxy  to  be  described  presently. 

(795.)  A  similar  course  of  observation  followed  out  in  the  southern 
hemisphere,  leads  independently  to  the  same  conclusion  as  to  the  law  of 
the  visible  distribution  of  stars  over  the  southern  galactic  hemisphere,  or 
that  half  of  the  celestial  surface  which  has  the  south  galactic  pole  for  its 
centre.    A  system  of  gauges,  extending  over  the  whole  surface  of  that 


'r''  (■ 


i:i  1li 


.IV)         !• 


<  I 


452 


OUTLINES  or  ^"T.  ONOMY. 


Zone*  of  Otiaetio  South 
Polar  Dlitanor. 

0" 

tol5« 

15 

to  80 

30 

to  45 

45 

to  60 

60 

to  75 

75 

to  90 

bcmisphero  takon  with  the  same  telcsoopo,  field  of  view  and  magnifying 
power  employed  in  Sir  William  Hersohers  gauges,  has  afforded  the  ave- 
rage numbers  of  stars  per  field  of  15'  in  diameter,  within  the  areas  of 
zones  enoiroling  that  polo  at  intervals  of  15°,  set  down  in  the  fuUuwing 
table. 

Arerag*  Number  of  Staia 
per  Fluid  of  liy. 

605 

6-62 

9-08 

13-49 

26-29 

69  06 

(796.)  These  numbers  are  not  directly  comparable  with  those  of  M. 
Struve,  given  in  Art  798,  because  the  latter  corresponds  to  the  limiting 
polar  distances,  while  these  are  the  averages  for  the  included  zones.  That 
eminent  astronomer,  however,  has  given  a  table  of  the  average  gauges  ap- 
propriate to  each  degree  of  north  galactic  polar  distance,'  from  which  it  is 
easy  to  calculate  averages  for  the  whole  extent  of  each  zone.  How  near 
a  parallel  the  results  of  this  calculation  for  the  northern  hemisphere  ex- 
hibit with  those  above  stated  for  the  southern,  will  be  seen  by  the  follow- 
ing table. 

ZouM  of  Qalaetle  North  ATornite  Number  of  Stan  per  Field 

Folar  DIstaiice.  of  ly  Arom  M.  StruTu's  Table. 

O"*  to  15°  4.32 

15   to  30  5-42 

30   to  45  8-21 

45   to  60  13-61 

60   to  75  2409 

75   to  90  53-43 

It  would  appear  from  this  that,  with  an  almost  exactly  similar  law  of  ap- 
parent density  in  the  two  hemispheres,  the  southern  were  somewhat  richer 
in  stars  than  the  northern,  which  may,  and  not  improbably  does,  arise 
r'rom  our  situation  not  being  precisely  in  the  middle  of  its  thickness,  but 
Komowhat  nearer  to  its  northern  surface. 

(797.)  When  examined  with  powerful  telescopes,  the  constitution  of 
this  wonderful  zone  is  found  to  be  no  less  various  than  its  aspect  to  the 
naked  eye  is  irregular.  In  some  regions  the  stars  of  which  it  is  wholly 
composed  are  scattered  with  remarkable  uniformity  over  immense  tracts, 
while  in  others  the  irregularity  of  their  distribution  is  quite  as  striking, 

*  Etudes  d'ABtronomie  Stellaire,  p.  34. 


TELESCOPIC  C0N8TITUTI0W   OP  TUB   GALAXY. 


468 


exhibiting  a  rapid  succession  of  closely  olustcring  rich  patches  separated 
by  comparatively  poor  intervals,  and  indeed  in  some  instances  by  spaces 
absolutely  dark  and  completdij  void  of  any  ttar,  even  of  the  smallest 
telescopic  magnitude.     In  some  places  not  more  than  40  or  60  stars  on 
au  average  occur  in  a  "gauge"  field  of  15',  while  in  others  a  similar 
average  gives  a  result  of  400  or  500.     Nor  is  less  variety  observable 
in  the  character  of  its  different  regions  in  respect  of  the  magnitudes  of 
the  stars  they  exhibit,  and  the  proportional  numbers  of  the  larger  and 
smaller  magnitudes  associated  together,  than  in  respect  of  their  aggregate 
numbers.     In  some,  for  instance,  extremely  minute  stars,  though  never 
altogether  wanting,  occur  in  numbers  so  moderate  as  to  lead  us  irresistibly 
to  the  conclusion  that  in  these  regions  we  see  fairly  through  the  starry 
stratum,  since  it  is  impossible  otherwise  (supposing  their  light  not  inter- 
cepted) that  the  numbers  of  the  smaller  magnitudes  should  not  go  on 
continually  increasing  ad  infinitum.     In  such  cases  moreover  the  ground 
of  tho  heavens,  as  seen  between  the  stars,  is  for  the  most  part  perfectly 
dark,  which  again  would  not  bo  the  case,  if  innumerable  multitudes  of 
stars,  too  minuto  to  be  individually  discernible,  existed   beyond.     In 
other  regions  wo  are  presented  with  the  phenomenon  of  an  almost  uni- 
form degree  of  brightness  of  tho  individual  stars,  accompanied  with  a 
very  even  distribution  of  them  over  the  ground  of  the  heavens,  both  tho 
larger  and  smaller  magnitudes  being  strikingly  deficient.     In  such  cases 
it  is  equally  impossible  not  to  perceive  that  we  are  looking  tlirougJi  a  sheet 
of  stars  nearly  of  a  size,  and  of  no  great  thickness  compared  with  the  dis- 
tance which  separates  them  from  us.     Were  it  otherwise  we  should  bo 
driven  to  suppose  the  more  distant  stars  uniformly  the  larger,  so  as  to 
compensate  by  their  greater  intrinsic  brightness  for  their  greater  distance, 
a  supposition  contrary  to  all  probability.     In  others  again,  and  that  not 
unfrequently,  we  are  presented  with  a  double  pheenomenon  of  the  same 
kind,  viz.  a  tissue  as  it  were  of  large  stars  spread  over  another  of  very 
small  ones,  the  intermediate  magnitudes  being  wanting.     The  conclusion 
here  seems  equally  evident  that  in  such  cases  we  look  through  two  side- 
real sheets  separated  by  a  starless  interval. 

(708.)  Throughout  by  far  the  larger  portion  of  the  extent  of  the  Milky 
Way  in  both  hemispheres,  the  general  blackness  of  the  ground  of  the 
heavens  on  which  its  stars  are  projected,  and  the  absence  of  that  innu- 
merable multitude  and  excessive  crowding  of  the  smallest  visible  magni- 
tudes, and  of  glare  produced  by  the  aggregate  light  of  multitudes  too 
small  to  affect  the  eye  singly,  which  the  contrary  supposition  would  appear 
to  necessitate,  must,  we  think,  be  considered  unequivocal  indications  that 
its  dimensions  in  directions  where  these  conditions  ohtaiUf  are  not  only  not 


=•1 


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454 


OUTLINES   OF   ASTRONOMY. 


infinite,  but  that  the  space-penefrating  power  of  our  telescopes  suffices 
fairly  to  piei'ce  through  and  beyond  it.  It  is  but  right,  however,  to  warn 
our  readers  that  this  conclusion  has  been  controverted,  and  that  by  an 
authority  not  lightly  to  be  put  aside,  on  the  ground  of  certain  views  taken 
by  Olbers  as  to  a  defect  of  perfect  transparency  in  the  celestial  spaces,  in 
virtue  of  which  the  light  of  the  more  distant  stars  is  enfeebled  more  than 
in  proportion  to  their  distance.  The  extinction  of  light  thus  originating, 
proceeding  in  geometrical  progression  while  the  distanoe  increases  in 
arithmetical,  a  limit,  it  is  argued,  is  placed  to  the  space-pc.iotrating  powers 
of  telescopes,  far  within  that  which  distance  alone  apart  from  such  obscu- 
ration would  assign.  It  would  lead  us  too  far  aside  of  the  objects  of  a 
treatise  of  this  nature  to  enter  upon  any  discussion  of  the  grounds  (partly 
metaphysical)  on  which  these  views  rely.  It  must  suffice  here  to  observe 
that  the  objection  alluded  to,  if  applicable  to  any,  is  equally  so  to  every 
part  of  the  galaxy.  We  are  not  at  liberty  to  argue  that  at  one  part  of  its 
circumference,  our  view  is  limited  by  this  sort  of  cosmical  veil  which 
extinguishes  the  smaller  magnitudes,  cuts  off  the  nebulous  light  of  distant 
masses,  and  closes  our  view  in  impenetrable  darkness;  while  at  another 
we  are  compelled  by  the  clearest  evidence  telescopes  can  afford  to  believe 
that  star-strown  vistas  lie  open,  exhausting  their  powers  and  stretching 
out  beyond  their  utmost  reach,  as  is  proved  by  that  very  phoenomenon 
which  the  existence  of  such  a  veil  would  render  impossible,  viz.  infinite 
increase  of  number  and  diminution  of  magnitude,  terminating  in  complete 
irresolvable  nebulosity.  Such  is,  in  effect,  the  spectacle  afforded  by  a  very 
large  portion  of  the  Milky  Way  in  that  interesting  region  near  its  point 
of  bifurcation  in  Scorpio  (arts.  789,  792,)  where,  through  the  hollows 
and  deep  recesses  of  its  complicated  structure  we  behold  what  has  all  the 
appearance  of  a  wide  and  indefinitely  prolonged  area  strewed  over  with 
discontinuous  masses  and  clouds  of  stars  which  the  telescope  at  length 
refuses  to  analyse.'  Whatever  other  conclusions  we  may  draw,  this  must 
any  how  be  regarded  as  the  direction  of  the  greatest  linear  extension  of 
the  ground-plaa  of  the  galaxy.  And  it  would  appear  to  follow,  also,  as  a 
not  less  obvious  consequence,  that  in  those  regions  where  that  zone  ia 
clearly  resolved  into  stars  well  separated  and  seen  projected  on  a  black 
ground,  and  where  by  consequence  it  is  certain  if  the  foregoing  views  be 


•  It  would  be  doing  great  injustice  to  the  illustrious  astronomer  of  Pulkova  (whose 
opinion,  if  we  hero  seem  to  controvert,  it  is  with  the  utmost  possible  defercr.ce  and 
respoci)  not  to  mention  that  ni  ilic  time  of  his  writing  tho  remarknlile  essny  already 
more  than  once  cited,  in  which  the  views  in  question  are  delivered,  he  could  not  havu 
been  aware  of  the  important  facts  alluded  to  in  the  text,  the  work  in  which  they  are 
described  being  then  unpublished. 


DISTANCE   OF  THE  FIXED   STARS. 


455 


correct  that  we  look  out  beyond  them  into  space,  the  smallest  visible  stars 
appear  as  such,  not  by  reason  of  excessive  distance,  but  of  a  real  inferiority 
uf  size  or  brightness. 

(799.)  When  we  speak  of  the  comparative  remoteness  of  certain 
regions  of  the  starry  heavens  beyond  others,  and  of  our  own  situation  in 
them,  the  question  immediately  arises,  what  is  the  distance  of  the  nearest 
fixed  star  ?  What  is  the  scale  on  which  our  visible  firmament  is  con- 
structed ?  And  what  proportion  do  its  dimensions  bear  to  those  of  our 
own  immediate  system?  To  these  questions  astronomy  has  at  length 
been  enabled  to  afford  an  answer. 

(800.)  The  diameter  of  the  earth  has  served  us  for  the  base  of  a  tri- 
angle, in  the  trigonometrical  survey  of  our  system  (art.  274,)  by  which  to 
calculate  the  distance  of  the  sun ;  but  the  extreme  minuteness  of  the  sun's 
parallax  (art.  357,)  renders  the  calculation  from  this  "ill-conditioned" 
triangle  (art.  275,)  so  delicate,  that  nothing  but  the  fortunate  combination 
of  favourable  circumstances,  afforded  by  the  transits  of  Venus  (art.  479,) 
could  render  its  results  even  tolerably  worthy  of  reliance.  But  the  earth's 
diameter  is  too  small  a  base  for  direct  triangulation  to  the  verge  even  of 
our  own  system  (art.  526,)  and  we  are,  therefore  obliged,  to  substitute 
the  annual  parallax  for  the  diurnal,  or,  which  comes  to  the  same  thing, 
to  ground  our  calculation  on  the  relative  velocities  of  the  earth  and  planets 
in  their  orbits  (art.  486,)  when  we  would  push  our  triangulation  to  that 
extent.  It  might  be  naturally  enough  expected,  that  by  this  enlargement 
of  our  base  to  the  vast  diameter  of  the  earth's  orbit,  the  next  step  in  our 
survey  (art.  275,)  would  be  made  at  a  great  advantage ; — that  our  change 
of  station,  from  side  to  side  of  it,  would  produce  a  considerable  and  easily 
measurable  amount  of  annual  parallax  in  the  stars,  and  that  by  its  means 
we  should  come  to  a  knowledge  of  their  distance.  But,  after  exhausting 
every  refinement  of  observation,  astronomers  were,  up  to  a  very  late  period, 
unable  to  come  to  any  positive  and  coincident  conclusion  upon  this  head ; 
and  the  amount  of  such  parallax,  even  for  the  nearest  fixed  star  examined 
with  the-  requisite  attention,  remained  mixed  up  with,  and  concealed 
among,  the  errors  incidental  to  all  astronomical  determinations.  The 
nature  of  these  errors  has  been  explained  in  the  earlier  part  of  this  work, 
and  we  need  not  lemind  the  reader  of  the  difficulties  which  must  necessa- 
rily attend  the  attempt  to  disentangle  an  element  not  exceeding  a  few 
tenths  of  a  second,  or  at  most  a  whole  second,  from  the  host  of  uncertain- 
ties entailed  on  the  results  of  observations  by  them  :  none  of  them  indi- 
vidually perhaps  of  greater  magnitude,  but  embarrassing  by  their  numbei 
and  fluctuating  amount.  Nevertheless,  by  successive  refinements  in 
instrument  making,  and  by  constantly  progressive  approximation  to  the 


I. 


r. 


t    !i 


•v 


456 


OUTLINES   OF  -ASTRONOMY. 


exact  knowledge  of  the  Uranographical  corrections,  that  assurance  had 
been  obtained,  even  in  the  earlier  years  of  the  present  century,  viz.  that 
no  star  visible  in  northern  latitudes,  to  which  attention  had  been  directed, 
manifested  an  amount  of  parallax  exceeding  a  single  second  of  arc.  It  is 
worth  while  to  pause  for  a  moment,  to  consider  what  conclusions  would 
follow  from  the  admission  of  a  parallax  to  this  amount. 

(801.)  Radius  is  to  the  sine  of  1"  as  206265  to  1.  In  this  proportion 
then  at  least  must  the  distance  of  the  fixed  stars  from  the  sun  exceed  that 
of  the  sun  from  the  earth.  Again,  the  latter  distance,  as  we  have  already 
seen  (art.  357,)  exceeds  the  earth's  radius  in  the  proportion  of  23984  to 
1.  Taking  therefore  the  earth's  radius  for  unity,  a  parallax  of  1"  sup- 
poses a  distance  of  4947059760  or  nearly  five  thousand  millions  of  such 
units :  and  lastly,  to  descend  to  ordinary  standards,  since  the  earth's  radius 
may  be  taken  at  4000  of  our  miles,  we  find  19788239040000  or  about 
twenty  billions  of  miles  for  our  resulting  distance. 

(802.)  In  such  numbers  the  imagination  is  lost.  The  only  mode  we 
have  of  conceiving  such  intervals  at  all  is  by  the  time  which  it  would 
require  for  light  to  traverse  them.  Light,  as  we  know  (art.  545,)  travels 
at  the  rate  of  192000  miles  per  second,  traversing  a  semidiameter  of  the 
earth's  orbit  in  8"  13''3.  It  would,  therefore,  occupy  206265  times  this 
interval,  or  3  years  and  83  days  to  traverse  the  distance  in  question.  Now 
as  this  is  an  inferior  limit  which  it  is  already  ascertained  that  even  the 
brightest  and  therefore  (in  the  absence  of  all  other  indications)  the  dis- 
tance of  those  innumerable  stars  of  the  smaller  magnitudes  which  the 
telescope  discloses  to  us  !  What  for  the  dimensions  of  the  gahixy  in 
whose  remoter  regions,  as  we  have  seen,  the  united  lustre  of  myriads 
of  stars  is  perceptible  only  in  powerful  telescopes  as  a  feeble  nebulous 
gleam  ! 

(803.)  The  space-penetrating  power  of  a  telescope,  or  the  comparative 
distance  to  which  a  given  star  would  require  to  be  removed,  id  order  that 
it  may  appear  of  the  same  brightness  in  the  telescope  as  before  to  the 
naked  eye,  may  be  calcuhited  from  the  aperture  of  the  telescope  cou) pared 
with  that  of  the  pupil  of  the  eye,  and  from  its  reflcctiog  or  transmitting 
power,  /.  e.  the  proportion  of  the  incident  light  it  convoys  to  the  observer's 
eye.  Thus  it  has  been  computed  that  the  space-penetrating  power  uf  such 
a  reflector  as  that  used  in  the  star-gauges  above  referred  to  is  expressed 
by  the  number  75.  A  star,  then,  of  the  sixth  magnitude,  removed  to 
75  times  its  distance,  would  still  be  perceptible  as  a  s/ar  with  that  instru- 
ment, and  admitting  such  a  star  to  have  100th  part  of  the  light  of  a 
standard  star  of  the  first  magnitude,  it  will  follow  that  such  a  standard 
star,  if  removed  to  750  times  its  distance,  would  expite  in  the  eye,  when 


DISTANCE   OF   THE   FIXED   STARS. 


467 


viewed  through  the  gauging  telescope,  the  same  laipression  as  a  star  of 
the  sixth  magnitude  does  to  the  naked  eye.  Among  the  infinite  multitude 
of  such  stars  in  the  remoter  regions  of  the  galaxy,  it  is  but  fair  to  con- 
clude that  innumerable  individuals  equal  in  intrinsic  brightness  to  those 
which  immediately  surround  us,  must  exist.     The  light  uf  such  stars, 
then,  must  have  occupied  upwards  of  2000  years  in  travelling  over  the 
distance  which  separates  them  from  our  own  system.     It  follows,  then, 
that  when  we  observe  the  places  and  note  the  appearance  of  such  stars,  we 
are  only  reading  their  history  of  two  thousand  years'  anterior  date,  thus 
wonderfully  recorded.   We  cannot  escape  this  conclusion,  but  by  adopting 
as  an  alternative  an  intrinsic  inferiority  of  light  in  all  the  smaller  stars 
of  the  galaxy.    "We  shall  be  better  able  to  estimate  the  probability  of  this 
alternative  when  we  shall  have  made  acquaintance  with  other  sidereal 
systems,  whose  existence  the  telescope  discloses  to  us,  and  whose  analogy 
will  satisfy  us  that  the  view  of  the  subject  here  taken  is  in  perfect  har- 
mony with  the  general  tenor  of  astronomical  facts. 

(804.)  Hitherto  we  have  spoken  of  a  parallax  of  1"  as  a  mere  limit, 
below  which  that  of  any  star  yet  examined,  assuredly,  or  at  least  very 
probably,  falls,  and  it  is  not  without  a  certain  convenience  to  regard  this 
amount  of  parallax  as  a  sort  of  unit  of  reference,  which,  connected  in  the 
reader's  recollection  with  a  parallactic  unit  of  distance  from  our  system 
of  20  billions  of  miles,  and  with  a  3i  years'  journey  of  light,  may  save 
him  the  trouble  of  such  calculations,  and  ourselves  the  necessity  of  <,over- 
ing  our  pages  with  such  enormous  numbers,  when  speaking  of  stars  whose 
parallax  has  actually  been  ascertahi:  d  with  some  approach  to  certainty, 
either  by  direct  meridian  observati,  j.,  or  by  more  refined  and  delicate 
methods.     These  we  shall  proceed  to  explain,  after  first  pointing  out  the 
theoretical  peculiarities  which  enable  us  to  separate  and  disentangle  its 
effects  from  those  of  the  Urair^irraphical  corrections,  and  from  other  causes 
of  error,  which,  being  periodical  in  their  nature,  add  greatly  to  the  diffi- 
culty of  the  subject.     The  eifects  of  precession  and  proper  motion  (see 
art.  852,)  which  are  uniformly  progressive  from  year  to  year,  and  that  of 
nutation,  which  runs  through  its  period  in  nineteen  years,  it  is  obvious 
enough,  separate  themselves  at  once  by  these  characters  from  that  of  pa- 
rallax ;  and,  being  known  with  very  great  precision,  and  being  certainly 
independent,  as  regards  their  causes,  of  any  incr/idual  peculiarity  in  the 
stars  affected  by  them,  whatever  small  uncertainty  may  remain  respecting 
the  numerical  elements  which  enter  into  their  computation  (or,  in  mathe- 
matical language,  their  co-rjicients),  can  give  rise  to  no  embarrassment. 
With  regard  to  aberration  the  case  is  materially  different.     This  correo 
tion  affects  the  place  of  a  star  by  a  fluctuation,  annual  in  its  period,  and 


sr,.. 


)  I 


'y^r 


458 


OUTLINES  OP  ASTRONOMY. 


therefore,  so  far  agreeing  with  parallax.  It  is  also  very  similar  in  tlio 
law  of  its  variation  at  diiFerent  seasons  of  the  year,  parallax  having  for 
its  apex  (see  art.  343,  344,)  the  apparent  place  of  the  sun  in  the  ecliptic, 
and  aberration  a  point  in  the  same  great  circle  90°  behind  that  place,  so 
that  in  fact  the  formulae  of  calculation  (the  coeflScients  excepted)  are  the 
same  for  both,  substituting  only  for  the  sun's  1  ugitude  in  the  expression 
for  the  one,  that  longitude  diminished  by  90°  for  the  other.  Moreover,  in 
the  absence  of  absolute  certainty  respecting  the  nature  of  the  propagation 
of  light,  astronomers  have  hitherto  considered  it  necessary  to  assume,  at 
least  as  a  possihility,  that  the  velocity  of  light  may  be  to  some  slight 
amount  dependent  on  individual  peculiarities  in  the  body  emitting  it.' 

(805.)  If  we  suppose  a  line  drawn  from  the  star  to  the  earth  at  all 
seasons  of  the  year,  it  is  evident  that  this  line  will  sweep  over  the  surfacu 
of  an  exceedingly  acute,  oblique  cone,  having  for  its  axis  the  line  joining 
the  sun  and  star,  and  for  its  base  the  earth's  annual  orbit,  which,  for  the 
present  purpose,  we  may  suppose  circular.  The  star  will  therefore  appear 
to  describe  each  year  about  its  mean  place,  regarded  as  fixed,  and  in  virtue 
of  parallax  alone,  a  minute  ellipse,  the  section  of  this  cone  by  the  surface 
Df  the  celestial  sphere,  perpendicular  to  the  visual  ray.  But  there  is  also 
another  way  iu  which  the  same  fact  may  be  represented.  The  apparent 
orbit  of  the  star  about  its  mean  place  as  a  centre,  will  be  precisely  that 
which  it  would  appear  to  describe,  if  seen  from  the  sun,  supposing  it 
really  revolved  about  that  place  in  a  circle  exactly  equal  to  the  earth's 
annual  orbit,  in  a  plane  parallel  to  the  ecliptic.  This  is  evident  from  the 
equality  and  parallelism  of  the  lines  and  directions  concerned.  Now  the 
effect  of  aborration  (disregarding  the  slight  variation  of  the  earth's  ve- 
locity in  different  parts  of  its  orbit)  is  precisely  similar  in  law,  and  difTers 
only  in  amount,  and  in  its  bearing  reference  to  a  direction  90°  different 
in  longitude.  Suppose,  in  order  to  fix  our  ideas,  the  maximum  of  parallax 
to  be  1"  and  that  of  aberration  20'5",  and  let  A  B,  ah,  be  two  circles 
imagined  to  be  described  separately,  as  above,  by  the  star  about  its  mean  place 
S,  in  virtue  of  these  two  causes  respectively,  S  'V  being  a  line  parallel  to  that 
of  thr  line  of  equinoxes.  Then,  if  ii'  virtue  of  parallax  alone,  the  star 
would  be  found  at  a  in  the  smaller  orbit,  it  would,  in  virtue  of  aberration 
alon«,  be  found  at  A,  in  the  larger,  the  angle  a  S  A  being  a  right  angle. 
^rawing  then  A  0  equal  and  parallel  to  S  a,  and  joining  S  C,  it  will  in 

'  In  the  actual  state  oi  astrononty  and  phototojry  this  necessity  can  hardly  be  consi- 
dered 08  sfil!  existinjf,  arid  it  w  dfuw^le,  fit^.fefore,  thai  the  practice  of  astronomers 
ot  introdii.  ,!ig  an  unknown  correwtori  ior  the  instant  of  aberration  into  their  "  equa- 
tions of  '  oiidiiion"  ft*  the  deteri»(M>«<ion  of  pfM^ti.B*,  should  be  disused,  since  it  actu- 
all*'  t«>n«*»  -v  mtroduo*  '/ror  into  n*»  f^ntX  reuuk. 


PARALLAX  OF  THE  FIXED  STARS. 


45d 


virtue  of  both  simultaneously  be  found  in  C,  i.  e.  in  the  circumference  of 
a  circle  whose  radiufj  is  S  C,  and  at  a  point  in  that  circle  in  advance  of  A, 
the  aberrational  pla-jo,  by  the  angle  A  S  C.  Now,  since  S  A  :  A  C  : : 
20'6  : 1,  we  find  for  the  angle  A  S  C  2"  47'  35",  and  for  the  length  of 

Fig.  lOit. 


the  radius  S  C  of  the  circle  representing  the  compound  motion  20''  524. 
The  difference  (0"-024)  between  this  and  S  C,  the  radius  of  the  aberra- 
tion circle,  is  quite  imperceptible,  and  even  supposing  a  quantity  so  mi- 
nute to  be;  capable  of  detection  by  a  prolonged  series  of  observations,  it 
would  remain  a  question  whether  it  were  produced  by  parallax  or  by  a 
specific  difference  of  aberration  from  the  general  average  20"-5  in  the  star 
itself.  It  is  therefore  to  the  difference  of  2°  48'  between  the  angular 
situation  of  the  displaced  star  in  this  hypothetical  orbit,  ^.  e.  in  the 
arguments  (as  they  are  called)  of  the  joint  correction  QV  S  C)  and  that  of 
aberration  alone  (T  S  A),  that  we  have  to  look  for  the  resolution  of  the 
problem  of  parallax.  The  reader  may  easily  figure  to  himself  the  deli- 
cacy of  an  inquiry  which  turns  wholly  (even  when  stripped  of  all  its  other 
difficulties)  on  the  precise  determination  of  a  quantity  of  this  nature,  and 
of  such  very  moderate  magnitude. 

(806.)  But  these  other  difficulties  themselves  are  of  no  trifling  order. 
All  astronomical  instruments  are  affected*  by  differences  of  temperature. 
Not  only  do  the  materials  of  which  they  are  composed  expand  and  con- 
tract, but  the  masonry  and  solid  piers  on  which  they  are  erected,  nay  even 
the  very  soil  on  which  these  are  founded,  participate  in  the  general  change 
.from  summer  warmth  to  winter  cold.  Hence  arise  slow  oscillatory  move- 
ments of  exceedingly  minute  amount,  which  levels  and  plumb-lines  afford 


(I'ii 


nil' 


\'-i\' 


'■f4^s 

m 


••«'• 


\'J- 


m^ 


■  ri' 


I;. 


460 


OUTLINES   OF  ASTRONOMY. 


;  js 


but  very  inadequate  means  of  detecting,  and  which  being  also  annual  in 
their  period  (after  rejecting  whatever  is  merely  casual  and  momentary) 
mix  themselves  intima'^ely  with  the  matter  of  our  inquiry.  Kefraction 
too,  besides  its  casual  variations  from  night  to  night,  which  a  long  scries 
of  observations  would  eliminate,  depends  for  its  theoretical  expression  on 
the  constitution  of  the  strata  of  our  atmosphere,  and  the  law  of  the  dis- 
tribution of  heat  and  moisture  at  different  elevations,  which  cannot  bo 
unaffected  by  difference  of  season.  No  wonder  then  that  mere  meridional 
observations  should,  almost  up  to  the  present  time,  have  proved  insuflBciont, 
except  in  one  very  remarkable  instance,  to  afford  unquestionable  evidence, 
and  satisfactory  quantitative  "leasurement  of  the  parallel  of  any  fixed 
star. 

(807.)  The  instance  referred  to  is  that  of  a  Centauri,  one  of  the  brightest 
and  for  many  other  reasons,  one  of  the  most  remarkable  of  the  southern 
stars.  From  a  series  of  observations  \'f  this  star,  made  at  the  Royal 
Observatory  of  the  Cape  of  Good  Hope  in  the  yenrs  1832  and  1833,  by 
Professor  Henderson,  with  the  mural  circle  of  that  ostablishment,  a  paral- 
lax to  the  amount  of  an  entire  second  was  conclude!  on  his  reduction  of 
the  observations  in  question  after  his  return  to  l!]ngland.  Subt;equent 
observations  by  Mr.  Maclear,  partly  with  the  same,  rnd  partly  with  a  new 
and  far  more  efficiently  constructed  instrument  of  the  same  description 
made  in  the  years  1839  and  1840,  have  fully  confirmed  the  reality  of  the 
parallax  indicated  by  Professor  Henderson's  observations,  though  with  a 
slight  diminution  in  its  concluded  amount,  which  comes  out  equal  to 
0".9128  or  about  jflths  of  a  second  j  hrujht  stars  in  its  immediate  nciijh- 
bourhood  being  unaffected  hi/  a  similar  periodical  displacement,  and  thus 
affording  mtisfactori/  proof  that  the  displacement  indicated  in  the  case 
of  the  star  in  question  is  not  merely  a  result  of  annual  variations  of 
temperature.  As  it  is  impossible  at  present  to  answer  for  so  minute  a 
quantity  as  that  by  which  this  result  differs  from  an  exact  second,  wo  may 
consider  the  distance  of  this  star  as  approximately  expressed  by  tho  j-'aral- 
\t:  ic  uii  I  of  distance  referred  to  in  art.  804. 

(808.)  A  short  ;  ue  previous  to  the  publication'  of  this  important 
result,  the  detection  of  a  sensible  and  measunible  amount  of  parallax  in  the 
star  N°  til  Cygni  of  Flamstcod's  catalogue  of  stars  was  anuouticed  by  the 
celebrated  astronomer  of  Kiinigsberg,  the  b'.te  M.  Bessel.^  This  is  a  small 
and  inconspicuous  star,  h.-irdly  excc'-ding  the  sixth  magnitude,  1  ut  which 
had  been  pointed  out  for  ospeciul  observation  by  the  ^eniurkablc  ciicum- 

'  Profpssor  Henderson's  paper  v^-i  read  before  the  Astt    [omical  Soci«ty  of  Loiidun, 
fan.  3,  1639.     Ii  Ivars  date  Dec.  -4,  lt^:!8. 
•Aatronomische  Nachrichten,  Nos.  365,  366.    Dec.  13,  1838. 


PARALLAX  OF  THE  FIXED  STABS. 


461 


>;  fi' 


stance  of  its  being  affected  by  Vi  proper  motion  (see  art.  852)  i.  e.  a  regular 
and  continually  progressive  annual  displacement  among  the  surrounding 
stars  to  the  extent  of  more  than  b"  per  annum,  a  quantity  so  very  much 
exceeding  the  average  of  similar  minute  annual  displacements  which  many 
other  stars  exhibit,  as  to  lead  to  a  suspicion  of  its  being  actually  nearer  to 
our  system.     It  is  not  a  little  remarkable  that  a  similar  presumption  of 
proximity  exists  also  in  the  case  of  a  Centauri,  whose  unusually  largo 
proper  motion  of  nearly  4"  per  annum  is  stated  by  Professor  Henderson 
to  have  been  the  motive  which  induced  him  to  subject  his  observations  of 
that  star  to  that  severe  discussion  which  led  to  the  detection  of  its  parallax. 
M.  Bessel's  observations  of  61  Cygni  were  commenced  in  August  1837, 
immediately  on  the  establishment  at  the  Konigsberg   observatory  of  a 
magnificent  heliometer,  the  workmanship  of  the  celebrated  optician  Fraun- 
hofer,  of  Munich,  an  instrument  especially  fitted  for  the  system  of  obser- 
vation adopted ;  which  being  totally  different  from  that  of  direct  meri- 
dional observation,  more  refined  in  its  conception,  and  susceptible  of  far 
greater  accuracy  in  its  practical  application,  we  must  now  explain. 

(809.)  Parallax,  proper  motion,  and  specific  aberration  (denoting  by 
the  latter  phrase  that  part  of  the  aberration  of  a  star's  light  which  may 
be  supposed  to  arise  from  its  individual  peculiarities,  and  which  we  have 
every  reason  to  believe  at  all  events  an  exceedingly  minute  fraction  of  the 
whole,)  are  the  only  uranograpbical  corrections  which  do  not  necessarily 
affect  alike  the  apparent  places  of  two  stars  situated  in,  or  veri/  nearly  in, 
tae  same  visual  line.  Supposing  then  two  stars  at  an  immense  distance, 
the  oiif  behind  the  other,  but  otherwise  so  situated  as  to  appear  very 
nearly  along  the  sar-  v'«ual  line,  they  will  constitute  what  is  called  a  star 
iyptlcally  double,  to  distinguish  it  from  a  star  physically  doublCf  of  which 
more  horeafter.  Aberration  (that  which  is  common  to  all  stars),  preces- 
sion, imtiition,  nay,  even  rrfradion,  and  instrumental  caiises  of  apparent 
diqJaeement,  mill  affect  them  alike,  or  so  very  nearly  alike  (if  the  minute 
difference  of  their  apparent  places  be  taken  into  account)  as  to  admit  of 
the  difference  being  neglected,  or  very  accurately  allowed  for,  by  an  easy 
calculation.  If  then,  instead  of  attempting  to  determine  by  observation 
the  pl*«*>  of  the  nearer  of  two  very  unequal  stars  (which  will  probably  be 
tiw  larger)  by  direct  observation  of  its  right  ascension  and  polar  distance, 
we  content  ourselves  with  referring  its  place  to  that  of  its  remoter  and 
smaller  companion  by  differential  observation,  i.  e.  by  measuring  only  its 
dlffermri'^  of  situation  from  the  httter,  we  are  at  once  relieved  of  the 
necessity  of  making  these  corrections,  and  from  all  uncertainty  as  to  their 
influence  on  the  result.  And  for  the  very  same  reason,  errors  of  adjust- 
ment (art.  136),  of  graduation,  and  a  host  of  instrumental  errors,  which 


;    H 


'^'i     -li^ 


^  Jills  i 


462 


OUTLINES    OF   ASTRONOMY. 


would  for  this  delicate  purpose  fatally  affect  the  absolute  determination  of 
either  star's  place,  are  harmless  when  only  the  difierence  of  their  places, 
each  equally  affected  by  such  causes,  is  required  to  be  known. 

(810.)  Throwing  aside  therefore  the  consideration  of  all  these  errors 
and  correotioDS;  and  disregarding  for  the  present  the  minute  effect  of 

Fig.  110. 


specific  aberration  and  the  uniformly  progressive  effect  of  proper  motion, 
let  us  trace  the  effect  of  the  differences  of  the  parallaxes  of  two  stars  thus 
juxtaposed,  or  their  apparent  relative  distance  and  position  at  various 
seasons  of  the  year.  Now  the  parallax  being  inversely  as  the  distance, 
the  dimensions  of  the  small  ellipses  apparently  described  (art.  805)  by 
each  star  on  the  concave  surface  (»f  the  heavens  by  parallactic  displacement 
will  differ, — the  nearer  star  describing  the  larger  ellipse.  But  both  stars 
lying  very  nearly  in  the  same  direction  from  the  sun,  these  ellipses  will  be 
similar  and  similarly  situated.  Suppose  S  and  s  to  be  the  positions  of  the 
two  stars  as  seen  from  the  sun,  and  let  A  B  C  D,  a  J  c  c?,  be  their  paral- 
lactic ellipses ;  then,  since  they  will  be  at  all  times  similarly  situated  in 
these  ellipses,  when  the  one  star  is  seen  at  A,  the  other  will  be  seen  at  a. 
When  the  earth  has  made  a  quarter  of  a  revolution  in  its  orbit,  their 
apparent  places  will  be  B  fe ;  when  another  quarter,  C  c ;  and  when 
another,  D  d.  If,  then,  we  measure  carefully,  with  micrometers  adapted 
for  the  purpose,  their  apparent  situation  with  respect  to  each  other,  at 
different  times  of  the  year,  we  should  perceive  a  periodical  change,  both 
in  the  direction  of  the  line  joining  them,  and  in  the  distance  between 
their  centres.  For  the  lines  A  a  and  C  c  cannot  be  parallel,  nor  the  lines 
B  h  and  D  d  equal,  unless  the  ellipses  be  of  equal  dimensions,  i.  e.  unless 
the  two  stars  have  the  same  parallax,  or  are  equidistant  from  the  earth. 
(811.)  Now,  micrometers,  properly  mounted,  enable  us  to  measure  very 


PARALLAX   OP   THE  FIXED   STARS. 


463 


exactly  both  the  distaoce  betwceu  two  objects  which  can  bo  soon  together 
in  the  same  field  of  a  telescope,  artd  the  position  of  the  line  joining  them 
with  respect  to  the  horizou,  or  the  mcridiany  or  any  other  determinate 
direction  in  the  heavens.  The  double  image  micrometer,  and  especially 
the  holiometer  (art.  200,  201)  is  peculiarly  adapted  for  this  purpose.  Tho 
images  of  tho  two  lars  formed  side  by  side,  or  in  the  same  line  prolonged, 
however  momentarily  displaced  by  temporary  refraction  or  instrumental 
tremor,  move  together,  preserving  their  relative  situation,  the  judgment 
of  which  is  no  way  disturbed  by  such  irrcular  movements.  The  helio- 
meter  also,  taking  in  a  greater  range  thari  ordinary  micrometers,  enables 
us  to  compare  one  large  star  with  more  than  one  adjacent  small  one,  and 
to  select  such  of  the  latter  among  many  near  it,  as  shall  be  most  favour- 
ably situated  for  the  detection  of  any  motion  in  the  large  one,  not  partici- 
pated in  by  its  neighbours. 

(812.)  The  star  examined  by  Bessel  has  two  such  neighbours,  both  very 
minute,  and  therefore  probably  very  distant,  most  favourably  situated,  the 
one  (s)  at  a  distance  of  7'  42",  the  other  («')  at  11'  46"  from  the  large 
star,  and  so  situated,  that  their  directions  from  that  star  mike  nearly  a 
right  angle  with  each  other.  The  effect  of  parallax  therefore  would 
necessarily  cause  the  two  distances  S  s  and  S  a'  to  vary  so  as  to  attain 
their  maximum  and  minimum  values  alternately  at  three-monthly  inter- 
vals, and  this  is  what  was  actually  observed  to  take  place,  the  one  distance 
being  always  on  the  increase  or  decrease  when  the  other  was  stationary 
(the  uniform  effect  of  proper  motion  being  understood  of  course  to  be 
always  duly  accounted  for).  This  alternation,  though  so  small  in  amount 
as  to  indicate,  as  a  final  result,  a  parallax,  or  rather  a  difference  of  paral- 
laxes between  the  large  and  small  stars  of  hardly  more  than  one  third  of 
a  second,  was  maintained  with  such  regularity  as  to  leave  no  room  for 
reasonable  doubt  as  to  its  cause,  and  having  been  confirmed  by  the  further 
continuance  of  these  observations,  and  quite  recently  by  the  exact  coinci- 
dence between  the  result  thus  obtained,  and  that  deduced  by  M.  Peters 
from  observations  of  the  same  star  at  the  observatory  of  Pulkova',  is  con- 
sidered on  all  hands  as  fully  established.  The  parallax  of  this  star  finally 
resulting  from  Bessel's  observation  is  0""348,  so  that  its  distance  from  our 
system  is  very  nearly  three  parallactic  units.  (Art.  804.) 

(813.)  The  bright  star  a  Lyrae  has  also  near  it,  at  only  43"  distance 
(and  therefore  within  the  reach  of  the  parallel  wire  or  ordinary  double 
image  micrometer)  a  very  minute  star,  which  has  been  subjected  since 
183.5  to  a  severe  and  assiduous  scrutiny  by  M.  Struve,  on  the  same  prin- 
ciple of  differential  observation.     He  has  thus  established  the  existence* 


m 


if 


'il 


^l  ! 


W'-ii 


Mm 


II 


«ii 


;- 


»  With  the  great  vertical  circle  by  Ertel. 


"f 


464  OUTLINES   OP  ASTKOiVOMY. 

of  a  mcaaural'Ic  amount  of  parallax  Id  the  larr^e  star,  IcsB  indocd  than 
♦hat  of  01  Cygni  (beiug  only  about  J  of  a  mccoikI),  but  yet  ?ufiiok'iit 
i^iuoh  wa8  the  delicacy  of  his  mep>Si<;r«mout«)  to  justify  this  excellent 
observer  iu  anuouucing  the  nvsult  m  nt  least  highly  probable,  on  the 
strength  of  only  fivfi  nights'  observntion,  ia  1835  aud  18i{6.  This  pro- 
bability, the  continuation  of  the  measures  to  the  end  of  1838  and  tho 
corroborative,  though  not  in  this  case  precisely  coincident,  result  of  Mr. 
Peters's  investigations  have  converted  into  certainty.  M.  St'-uvc  has 
the  merit  of  being  the  first  to  bring  into  pra  "al  application  tiiist  method 
of  observation,  which,  though  propoi5cd  for  \  e  purpose,  aud  its  grcist  ml- 
vantages  pointed  out  by  Sir  William  Henschel  so  early  as  1781',  rcmuitiod 
long  unproductive  of  any  result,  owing  partly  to  the  imperfection  of 
nncromcters  for  the  measurement  of  distance,  and  partly  to  a  i-cason 
which  we  shall  presently  have  occasion  to  refer  to. 

(814.)  If  the  component  individuals  S,  s  (Jii/.  nxt.  810,)  bo  (as  is  often 
tho  case)  very  close  to  each  other,  tho  parallactic  variation  of  their  ani/k 
of  position,  or  the  extreme  angle  included  between  the  linesAa,  Cc, 
may  be  very  considerable,  even  for  a  small  amount  of  Jiiferenco  of  paral- 
laxes between  the  largo  and  small  stars.  For  instance  in  the  case  of  two 
adjacent  stars  15"  asunder,  and  otherwise  favourably  situated  for  observa- 
tion, an  annual  fluctuation  to  and  fro  iu  the  apparent  direction  of  their 
line  of  junction  to  the  extent  of  half  a  degree  (a  quantity  which  could 
not  escape  notice  in  the  means  of  numerous  and  careful  measurements) 
would  correspond  to  a  difference  of  parallax  of  only  I  of  a  second.  A 
difference  of  1"  between  two  stars  apparently  situated  at  5"  distance 
might  cause  an  oscillation  in  that  line  to  the  extent  of  no  Iqsh  than  IP, 
and  if  nearer  one  proportionally  still  greater.  This  mode  of  observation 
has  not  yet  been  put  in  practice,  but  seems  to  offer  great  advantages.* 

(815.)  The  following  is  a  list  of  stars  to  which  parallax  has  been  up 
to  the  present  time  more  or  less  probably  assigned  : 

a  Centauri    -        •• 0'913  (Henderson.) 

61  Cy./ni 0-348  (Besat!.) 

a  L\  tOB 0"26I  (Sinive.) 

Kiriiis 0'230(Heiider8on.) 

1830  (iroombndge' 0-226 (Peters.) 

UrsiB  Majoris  0'I33    ditto. 

Arcturus  0-127    ditto. 

Polaris  -        0067    ditto. 

Capella  0-046    ditto. 

*  It  has  been  referred  even  to  Galileo.  But  the  general  explanation  of  Parallax  in 
tiie  SyBtema  Cosmicum,  Dial.  iii.  p.  271  (Leyden  edit.  1699)  to  which  the  reference 
applies,  docs  not  touch  any  of  the  peculiar  features  of  the  cane,  or  meet  cny  of  its 
difficulties. 

*  Sec  Phil.  Trans.  1826.  p.  266,  et  seq.  and  1827,  for  a  list  of  stars  well  adapted  for 
«uch  observation,  with  the  times  of  the  year  most  favourable. — The  list  in  Phil.  Trcns. 
1826,  is  ii\correct. 

*  Groombridge's  catalogue  of  circumpolar  stars. 


PARALLAX  OF  THE  FIXED  STARS. 


465 


Although  the  extroino  nuauteness  of  the  la^t  four  of  these  rcHultu  de- 
prives tbc'iu  of  much  Duraerical  roliance,  it  is  at  least  certain  that  the 
parallaxes  by  no  nicjins  follow  the  i^rder  of  magnitudes,  and  this  is  farther 
shown  by  the  fact  that  u  Cygni,  one  of  M.  Peters's  stars,  shows  absolutely 
no  indications  of  any  measurable  parallax  whatever. 

(81G.)  From  the  distance  of  the  stars  we  arc  naturally  led  to  the  con- 
sideration of  their  real  magnitudes.  But  hero  a  difficulty  arises,  which, 
80  far  as  we  can  judge  of  what  optical  instruments  are  capable  of  cffi-ct- 
iug,  must  always  remain  inf  able.  Telescopes  afford  us  only  negative 
information  as  to  the  a, ;  ^'ular  diameter  of  any  star.     The  rotmd, 

well-defined,  planetary  umi  d  telescopes  show  when  turned  upon 

any  of  the  brighter  stars  u  ;ua  of  diffraction,  dependent,  though 

at  present  somewhat  enigm..  ically,  tu  the  mutual  interference  of  the  rays 
of  light.  They  are  consequently,  so  far  as  this  inquiry  is  concerned, 
more  optical  illusions,  and  have  therefore  been  termed  spurious  discs. 
The  proof  of  this  is  that  telescopes  of  different  apertures  and  magnifying 
powers,  when  applied  for  the  purpose  of  measuring  their  angular  diame- 
ters, give  differont  rcisults,  the  greater  aperture  (even  with  the  same  mag- 
nifying power  giving  the  smaller  disc.  That  the  true  disc  of  even  a 
large  and  bright  star  can  have  but  a  very  minui!.-  angular  measure,  ap- 
pears from  the  fact  that  in  the  occultation  of  such  a  star  by  the  moon,  Its 
extinction  is  ahsoluuli/  instantaneous,  not  the  smallest  trace  of  gradual 
diminution  of  light  being  perceptible.  The  apparent  or  spurious  disc 
also  remains  perfectly  round  and  of  Us  full  size  up  to  the  instant  of  dis- 
appearance, which  could  not  be  the  case  were  it  a  real  object.  If  our  suu 
wore  removed  to  the  distance  expressed  by  our  parallactic  unit  (art.  804), 
its  apparent  diameter  of  32'  3"  would  be  reduced  to  only  0"-0093,  or  lcs8 
than  the  hundredth  of  a  second,  a  quantity  which  wo  have  not  the  smallest 
reason  to  hope  any  practical  improvement  in  telescopes  will  ever  show  as 
an  object  having  distinguishable  form. 

(817.)  There  remains  therefore  only  the  indication  which  the  quantity 
of  light  they  send  to  us  may  afford.  But  here  again  another  difficulty 
besets  us.  The  light  of  the  sun  is  so  immensely  superior  in  intensity  to 
that  of  any  star,  that  it  is  impracticable  to  obtain  any  direct  comparison 
between  them.  But  by  using  the  moon  as  an  intermediate  term  of  com- 
parison it  may  be  done,  not  indeed  with  much  precision,  but  sufficiently 
well  to  satisfy  in  some  degree  our  curiosity  on  the  subject.  Now  a  Cen- 
tauri  has  been  directly  compared  with  the  moon  by  the  method  explained 
in  Art.  783,  By  a  mean  of  eleven  such  comparisons  made  in  various 
states  of  the  moon,  duly  reduced  and  making  the  proper  allowance  oa 
photometric  principles  for  the  moon's  light  lost  by  transmission  througfx 
30 


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IMAGE  EVALUATION 
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Photographic 

Sdences 

Corporation 


23  WEST  MAIN  STREET 

WEBSTER,  N.Y.  145*0 

(716)  872-4503 


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C^ 


466 


OUTLINES  OP  ASTRONOMY.  \T 


the  lens  and  prism,  it  appears  that  the  meaD  quantity  of  light  sent  to  the 
earth  by  a  full  moon  exceeds  that  sent  by  a  Centauri  in  the  proportion  of 
27408  to  1.  Now  Wollaston,  by  a  method  apparently  unobjectionable, 
found'  the  proportion  of  the  sun's  light  to  that  of  the  full  moon  to  be 
that  of  801072  to  1.  Combining  these  results,  we  find  the  light  sent  us 
by  the  sun  to  be  that  sent  by  •  Centauri  as  21,955,000,000,  or  about 
twenty-two  thousand  millions  to  1.  Hence  from  the  parallax  assigned, 
above  to  that  star,  it  is  easy  to  conclude  that  its  intrinsic  splendour,  as 
compared  with  that  of  our  sun  at  equal  distances,  is  2*8247,  that  of  the 
sun  being  unity.' 

(818.)  The  light  of  Sirir.a  is  four  times  that  of  a  Centauri  and  its  pa- 
rallax only  0"'280  (Art.  280).  This  in  effect  ascribes  to  it  an  intrinsic 
splendour  equal  to  68-02  times  that  of  our  sun.'      -  i,y  ;^'#ryt  ^, 

'  Wollaston,  Phil.  Trans.  1829,  p.  27.    ''^   •  .'  >' >^r 

*  BeBuUa  of  ABlronomieal  Ohservatiotu  at  the  Cape  of  Good  Hope,  <(«.  Art.  278,  p. 
363.  If  only  the  results  obtained  near  the  quadratures  of  the  moon  (which  ia  the  sit- 
uation inodt  favourable  to  exactness)  be  used,  the  resulting  value  of  the  intrinsic  light 
of  the  star  (the  sun  being  unity)  is  4*1586.  On  the  other  hand,  if  only  those  procured 
near  the  full  moon  (the  vrorst  time  for  observation)  be  employed,  the  result  is  1'4017. 
Discordances  of  this  kind  will  startle  no  one  conversant  with  Photometry.  That  a 
Centauri  really  emits  more  light  than  our  sun  must,  we  conceive,  be  regarded  as  an 
established  fact.  To  those  who  may  refer  to  the  work  cited  it  is  necessary  to  mention 
that  the  quantity  there  designated  by  M,  expresses,  on  the  scale  there  adopted,  500 
times  the  actual  illuminating  power  of  the  moon  at  the  time  of  observation,  that  of  the 
mean  full  moon  benig  unity. 

*  See  the  work  above  cited,  p.  367. — Wollaston  makes  the  light  of  Sirius  one  20,000- 
millionth  of  the  sun's.  Steinheil  by  a  very  uncertain  method  found  O  ax  (3286500)'X 
Arcturus. 


.'.J'.,- 


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PERIODICAL  STARS. 


467 


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i-;,>*  ^.-y^-.  -■.{ .,',  vi^  ^•.  ^i  ;■  CHAPTER  XVL 


t-   J. 


YARIABLB  AND  PERIODICAL  STABS. — LIST  OF  THOSE  ALREADT  KNOWN. 

—  IRREGULARITIES  IN  THEIR  PERIODS  AND  LUSTRE  WHEN  BRIGHT- 
EST.—  IRREGULAR  AND  TEMPORARY  STARS. —  ANCIENT  CHINESE  RE- 
CORDS OF  SEVERAL.  —  MISSING  STARS.  —  DOUBLE  STARS.  —  THEIR 
CLASSIFICATION. —  SPECIMENS  OF  EACH  CLASS. — BINARY   SYSTEMS. 

—  REVOLUTION  ROUND  EACH  OTHER.  —  DESCRIBE  ELLIPTIC  ORBITS 
UNDER  THE  NEWTONIAN  LAW  OP  GRAVITY. —  ELEMENTS  OF  ORBITS 
OP  SEVERAL. — ACTUAL  DIMENSIONS  OF  THEIR  ORBITS. —  COLOURED 
DOUBLE  STARS. — PH^SINOMENON  OF  COMPLEMENTARY  COLOURS. — 
SANGUINE  STARS. — PROPER  MOTION  OF  THE  STARS. — PARTLY  AC- 
COUNTED FOR  BY  A  REAL  MOTION  OF  THE  SUN.  —  SITUATION  OF 
THE  SOLAR  APEX. — AGREEMENT  OF  SOUTHERN  AND  NORTHERN 
STARS  IN  GIVING  THE  SAME  RESULT. — PRINCIPLES  ON  WHICH  THE 
INVESTIGATION  OF  THE  SOLAR  MOTION  DEPENDS. — ABSOLUTE  VE- 
LOCITY OF  THE  bun's  motion.  —  SUPPOSED  REVOLUTION  OF  THE 
WHOLE  SIDEREAL  SYSTEM  ROUND  A  COMMON  CENTRE. —  SYSTEMA- 
TIC PARALLAX  AND  ABERRATION. — EFFECT  OF  THE  MOTION  OF 
LIGHT  IN  ALTERING  THE  APPARENT  PERIOD  OF  A  BINARY  STAR. 


; 
I 

1 


(819.)  Now,  for  what  purpose  are  we  to  suppose  such  magnificent  bodies 
scattered  through  the  abyss  of  space?  Surely  not  to  illuminate  our 
nights,  which  an  additional  moon  of  the  thous;mdth  part  of  the  size  of 
our  own  would  do  much  better,  nor  to  sparkle  as  a  pageant  void  of  mean- 
ing and  reality,  and  bewilder  us  among  vain  conjectures.  Useful,  it  is 
true,  they  are  to  man  as  points  of  exact  and  permanent  reference ;  but  he 
luust  have  studied  astronomy  to  little  purpose,  who  can  suppose  man  to  be 
the  only  object  of  his  Creator's  care,  or  who  does  not  see  in  the  vast  and 
wonderful  apparatus  around  us  provision  for  other  races  of  animated 
beings.  The  planets,  as  we  have  seen,  derive  their  light  from  the  sun  j 
but  that  cannot  be  the  case  with  the  stars.  These  doubtless,  then,  are 
themselves  suns,  and  may,  perhaps,  each  in  its  sphere,  be  the  presiding 
centre  round  which  other  planets,  or  bodies  of  which  we  can  form  no  con- 
ception from  any  analogy  offered  by  our  own  system,  may  be  circulating. 


1 " ,' 


468 


OUTLINES  OF  ASTRONOMT. 


(820.)  Analogies,  however,  more  than  oonjectaral,  are  not  wanting  to 
indicate  a  correspondence  between  the  dynamical  laws  which  prevail  in  the 
remote  regions  of  the  stars  and  those  which  govern  the  motions  of  our  own 
system.  Wherever  we  can  trace  the  law  of  periodicity  —  the  regular  re- 
currence of  the  same  phsenomena  in  the  same  times — we  are  strongly 
impressed  with  the  idea  of  rotatory  or  orbitual  motion.  Among  the  stars 
are  several  which,  though  no  way  distinguishable  from  others  by  any  appa- 
rent change  of  place,  nor  by  any  difference  of  appearance  in  telescopes, 
yet  undergo  a  more  or  less  regular  periodical  increase  and  diminution  of 
lustre,  involving  in  one  or  two  cases  a  complete  extinction  and  revival. 
These  are  called  periodical  stars.  The  longest  known  and  one  of  the  most 
remarkable  is  the  star  Omicron,  in  the  constellation  Cetus  (sometimes 
called  Mira  Ceti),  which  was  first  noticed  as  variable  by  Fabricius  in  1596. 
It  appears  about  twelve  times  in  eleven  years,  or  more  exactly  in  a  period 
of  83P  IS''  7" ;  remains  at  its  greatest  brightness  about  a  fortnight,  being 
then  on  some  occasions  equal  to  a  large  star  of  the  second  magnitude ; 
decreases  during  about  three  months,  till  it  becomes  completely  invisible 
to  the  naked  eye,  in  which  state  it  remains  about  five  months :  and  con- 
tinues increasing  during  the  remainder  of  its  period.  Such  is  the  general 
course  of  its  phases.  It  does  not  always  however  return  to  the  same 
degree  of  brightness,  nor  increase  and  diminish  by  the  same  gradations, 
neither  are  the  successive  intervals  of  its  maxima  equal.  From  the  recent 
observations  and  inquiries  into  its  history  by  M.  Argelander,  the  mean 
period  above  assigned  would  appear  to  be  subject  to  a  cyclical  fluctua- 
tion embracing  eighty-eight  such  periods,  and  having  the  effect  of 
gradually  lengthening  and  shortening  alternately  those  intervals  to  the 
extent  of  twenty-five  days  one  way  an('  other.'  The  irregularitiea  in 
the  degree  of  brightness  attained  at  the  >^ximum  are  probably  also  peri- 
odical. Hevelius  relates'  that  during  the  four  years  between  October 
1672  and  December  1676  h  lid  not  appear  at  all.  It  was  unusually 
bright  on  October  5,  1839  (th«)  epoch  of  its  maximum  for  that  year  ac- 
cording to  M.  Argelander 's  observations)  when  it  exceeded  a  Ceti  and 
equalled  P  Aurig»  in  lu.<^tre. 

(821.)  Another  very  remarkable  periodical  star  is  that  called  Algol,  or 
p  Persei.  It  is  usually  visible  as  a  star  of  the  second  magnitude,  and 
quch  it  continues  for  the  space  of  2'  13^',  when  it  suddenly  begins  to  di- 
minish in  splendour,  and  in  about  8}  hours  is  reduced  to  the  fourth  mag- 
nitude, at  which  it  continues  about  15".  It  then  begins  again  to  increase, 
and  in  3 1  hours  more  is  restored  to  its  usual  brightness,  going  through  all 
its  changes  in  2*  20''  48»  58»'5.    This  remarkable  law  of  variation  cer- 

*  Astronom.  Nachr.  No.  624.  *  Lalande's  Astronomy,  Art.  794. 


X 


PERIODICAL  STARS,  fry 


469 


tainlj  appears  strongly  to  suggest  the  revolution  round  it  of  some  opaque 
body,  which,  when  interposed  between  us  and  Algol,  outs  off  a  large  por- 
tion of  its  light;  and  this  is  accordingly  the  view  taken  of  the  matter  by 
Goodricke,  to  whom  we  owe  the  discovery  of  this  remarkable  fact,'  in  the 
year  1782 ;  since  which  time  the  same  pheenonieua  have  continued  to  be 
observed,  but  with  this  remarkable  additional  point  of  interest,  viz.  that 
the  more  recent  observations,  as  compared  with  the  earlier  ones,  indicate 
a  diminution  in  the  periodic  time.  The  latest  observations  of  Argelander, 
Heis,  and  Schmidt,  even  go  to  prove  that  this  diminution  i;  not  uniformly 
progressive,  but  is  actually  proceeding  with  accelerated  rapidity,  which 
however  will  probably  not  continue,  but,  like  other  cyclical  combinations 
in  astronomy,  will  by  degrees  relax,  and  then  be  changed  into  an  increase, 
according  to  laws  of  periodicity  which,  as  well  as  their  causes,  remain  to 
be  discovered.  The  first  minimum  of  this  star  in  the  year  1844  occurred 
on  Jan.  3,  at  4*  14»  Greenwich  mean  time.' 

(822.)  The  star  j  in  the  constellation  Cepheus  is  also  subject  to  peri- 
odical variations,  which,  from  the  epoch  of  its  first  observation  by  Good- 
ricke in  1784  to  the  present  time,  have  been  continued  with  perfect  regu- 
larity. Its  period  from  minimum  to  minimum  is  5*  8''  47'"  39'-5,  tho 
first  or  epochal  minimum  for  1849  falling  on  Jan.  2,  3"  18»  37»  M.  T.  at 
Greenwich.  The  extent  of  its  variation  is  from  the  fifth  to  between  the 
third  and  foiirth  magnitudes.  Its  increase  is  more  rapid  than  its  diminu- 
tion, the  interval  between  the  minimum  and  maximum  of  its  light  being 
only  1*  14'',  while  that  from  the  maximum  to  the  minimum  is  3"  19". 

(823.)  The  periodical  star  p  Lyrae,  discovered  by  Goodricke  also  in 
1784,  has  a  period  which  has  been  usually  stated  at  from  6*  9''  to  6*  11*, 
and  there  is  no  doubt  that  in  about  this  interval  of  time  its  light  under- 
goes a  remarkable  diminution  and  recovery.  The  more  accurate  observa- 
tions of  M.  Argelander  however  have  led  him  to  conclude'  the  true  period 
to  be  12'  21"  dS"  10',  and  that  in  this  period  a  double  maximum  and 
minimum  takes  place,  the  two  maxima  being  nearly  equal  and  both  about 

'  The  same  discovery  appears  to  have  been  made  nearly  about  the  same  time  by 
Palitzcli,  a  farmer  of  Prolitz,  near  Dresden, — a  peasant  by  station,  au  astronomer  by 
nature, — who,  from  his  familiar  acquaintance  with  the  aspect  of  the  heavens,  had  been 
led  to  notice  among  so  many  thousand  stars  this  one  as  distinguished  from  the  rest  by  its 
variation,  and  had  ascertained  its  period.  The  same  Palitzch  was  also  the  first  to  re- 
discover the  predicted  comet  of  Halley  in  1759,  which  he  saw  nearly  a  month  before 
any  of  the  astronomers,  who,  armed  with  their  telescopes,  were  anxiously  watching 
its  return.    These  anecdotes  carry  us  back  to  the  era  of  the  Chaldean  shepherds. 

» Ast.  Nach.  No.  472. 

'  Astron.  Nachr.  No.  264.  See  also  the  valuable  papers  by  this  excellent  astron- 
omer in  A.  N.  Nos.  417,  455,  &c. 


«>l 


470 


OUTLINES  OF  ASTRONOMT. 


/ 


the  3*4  magnitade,  but  the  minima  considerably  unequal,  vis.  4*3  and 
4-5m.  In  addition  to  this  curious  subdivision  of  the  whole  interval  of 
change  into  two  semi-periods,  we  are  presented  in  the  case  of  this  star 
wivh  another  instance  of  slow  alteration  of  period,  which  has  all  the  a[K 
pearance  of  being  itself  periodical.  From  the  epoch  of  its  discovery  in 
1784  to  the  year  1840  the  period  was  continually  lengthening,  but  more 
and  more  slowly,  till  at  the  last-mentioned  epoch  it  ceased  to  increase,  and 
has  since  been  slowly  on  the  decrease.  As  an  epoch  for  the  least  or  ab- 
solute minimum  of  this  star,  M.  Argelander's  calculations  enable  us  to 
assign  1846  January  3«  0"  9-  63«  G.  M.  T. 

(824.)  Another  periodical  star  whose  changes  have  been  carefully  ob- 
served is  11  Aquilse  or  Antinoi,  first  pointed  out  by  Pigott  in  1784  (a  year 
fertile  in  such  discoveries)  as  belonging  to  that  class.  Its  period  is  7'  4'> 
13"  53*,  the  first  minimum  for  1849  occurring  on  Jan.  2,  at  19^  22"  55' 
Or.  M.  T.  It  occupies  fifty-seven  hours  in  its  increase  from  5m  to  4-8m, 
and  115  hours  in  its  decrease 

(825.)  These  are  all  the  variable  stars  which  have  been  observed  with 
sufficient  care  and  for  a  sufficient  length  of  time  to  enable  us  to  speak 
with  precision  as  to  their  periods,  epochs,  and  phases  of  brightness.  But 
the  number  of  those  whose  period  is  approximately  or  roughly  known  is 
considerable,  and  of  those  whose  change  is  certain,  though  its  period  and 
limits  are  as  yet  unknown,  still  more  so.  The  following  table  includes 
the  principal  among  them,  though  each  year  adds  to  their  number : — 


star. 


/J  Persei  (Algol) 

XTauri 

Cephei....   

t)  Aquilse 

*  Cancri  R.  A.  (1800)  =» 

8"  32"'-6  N.  P.  D.  70°  15' 

(  Oeminorum 

jihyrsB 

a  Herculis  

69  B.  Scuti  R.  A.  1801  = 

18"  37";  N.  P.  D.  =  960  57'.... 

c  AurigSB 

0  Cetl  (Mira) 

«  Serpen  tis  R.  A.  1828  = 

15"  46"  45';  P.  D.  74"»  20'  30" 

y  Cygni 

V  HydrsB  (B.  A.  0.  4501.) 

*  Cephei  (B.  A.  C.  7582.) 

34  Cygni  (B.  A.  C.  6990.) 

»  Leonis  (B.  A.  C.  3345.) 

<8agittarii 

t/r  Leonis 


Period. 


Change  of  Mag. 


d.    dec. 
2-8673 

*± 

6-36G4 

7-1763 

9015 
10-2 
12-9119 
63  ± 

71-200 
250  ± 
331-63 

336  ± 
396-875 

494  rb 
5  or  6  years 
18  years  ± 
Many  years 
Ditto 
Ditto 


from 

2 

4 

3-4 

3-4 

7-8 
4-a 
3-4 
3 

6 
3 
2 

7? 

6 

4 

3 

6 

6 

3 

6 


to 
4 

6-4 
6 
4'5 

10 
4-5 
4-6 
4 

0 

4 
0 

0 
11 
10 

6 

0 

0 

« 

0 


DlscoTered  by 


Goodricke,  1782. 
Baxendell,  1848. 
Goodricke,  1784. 
Pigott,  1784. 

Hind,  1848. 
Schmidt,  1847. 
Goodricke,  1784. 
Horschel,  1796. 

Pigott,  1795. 
I^eis,  1846. 
Fabricius,  1596. 

Harding,  1826. 
Kirch,  1687. 
Maraldi,  1704. 
Herschel,  1782. 
Janson,  1600. 
Koch,  1782. 
Halley,  1676. 
Montaoari,  1667. 


VARIABLE  AND  PERIODICAL  STARS. 


471 


Star. 


ijOygni 

*  Virginia  R.  A.  (1840)  = 

12"  3»;  N.  P.  D.  82°  8' 

*  CoronsB  Bor.  (B.  A.  C.  5236)..., 

TArietis  (B.  A.  0.681.) 

ij  Argfla 

a  Orionis  

a  UrsBB  Migorig 

i;  UrsBB  Majoris , 

^  Urssa  Minorii 

a  CassiopeiaB 

a  Hydrae 

*  R.  A.  (1847)  =  22"  68°  67-9  N. 
P.  D.  =  80<'  17' 30" 

*  R.  A.  (1848)  =  7"  33«  66-2  N. 
P.  D.-fle"  11'  66" 

*  R.  A.  (1848)  =■  7"  40»  10-3  N. 
P.  D.  =- 66°  63' 29" 

Near  *  R.  A.  22"  21m  0-4  (1848.) 
N.  P.  D.  100°  42'  40" 

*  R.  A.  (1848)  14"  44"  39'-6  N.  P. 
D.  101°  45' 26" 

i  Ursse  Migoris 


Period. 


d.    deo. 
Many  years 

146  days 

10}  months 

6  years? 

Irregular 

Ditto 

Some  years 

Ditto 

2  or  3  years? 

226  days  ? 

29  or  30  days? 

Unknown 

Ditto 

Ditto 

Ditto 

Ditto 
Many  years 


Change  of  Mag. 


from 
4-6 

6-7 

6 

A 

1 

1 

1-2 

1-2 

2 

2 

2-3 

8? 

9 

9 

7-S 

8 
2? 


to 
6-6 

0 

0 

8 

4 

1-2 

2 

2 

2-3 

2-3 

3 


0 

0 

0 

9-10 
2-3 


DiMorered  hT 


Herschel,jr.,1842? 

Harding,  1814. 
Pigott,  1796. 
Piazzi,  1798. 
Burchell,  1827. 
Herschel,  jr.,  1836. 
Ditto,  1846. 
Ditto,  1846. 
Struve,  1838. 
Herschel,  jr.,  1838. 
Ditto,  1837. 

Hind,  1848. 

Ditto,  1848. 

Ditto,  1848. 

RUmker. 

Schumacher. 
Matter  of  general 
remark. 


N.  B.  In  the  above  list  the  letters  B.  A.  C.  indicate  the  catalogue  of  the  British  Asso- 
ciation, B.  the  catalogue  of  Bode.  Numbers  before  the  name  of  the  constellation  (as 
34  Cygni)  denote  Flamsteed's  stars.  Since  this  table  was  drawn  up,  four  additional 
stars,  variable  from  the  8th  or  9th  magnitude  to  0,  have  been  communicated  to  us  by 
Mr.  Hind,  whose  places  are  as  follows:  (1.)  R.  A.  1"  38"  24«;  N.  P.  D.  81°  9'  39"'; 
(2.)  4"  50»  42",  82°  6'  36"  (1846) ;  (3.)  8"  43»  8.,  86°  11'  (1800) ;  (4.)  22"  12""  9',  82° 
59'  24"  (1800.)  Mr.  Hind  remarks  that  about  several  variable  stars  some  degree  of 
haziness  is  perceptible  at  their  minimum.  Have  they  clouds  revolving  round  them  as 
planetary  or  cometary  attendants?  He  also  draws  attention  to  the  fact  that  the  red 
colour  predominates  among  variable  stars  generally.  The  double  star.  No.  27] 8  of 
Struve's  Catalogue,  R.  A.  20"  34'°,  P.  D.  77°  54',  is  stated  by  the  author  to  be  variable. 
Captain  Smyth  (Celestial  Cycle,  i.  274)  mentions  also  3  Leonis  and  18  Leonis  aa 
variable,  the  former  from  6"  toO,  P  =  78  days,  the  latter  fi-om  5"  to  10",  P  =  311"'  23", 
but  without  citing  any  authority.  Piazzi  sets  down  96  and  97  Virginis  and  38  HercuUn 
as  variable  stars.  [The  blood-red  star,  4"  51"  50*9%  102°  2'  4"  (1850),  discovered  by 
Mr.  Hind,  is  stated  by  Schmidt  (Ast.  Nachr.  760)  to  have  been  seen  by  him  &^  in  Jan. 
1850,  and  to  have  totally  disappeared  in  Dec.  1850  and  Jan.  1851.] 

(826.)  Irregularities  similar  to  those  which  have  been  noticed  in  the 
case  of  0  Ceti,  in  respect  of  the  maxima  and  minima  of  brightness  attained 
m  successive  periods,  have  been  also  observed  in  several  others  of  the  stars 
in  the  foregoing  list,  x  Cygni,  for  example,  is  stated  by  Cassini  to  have 
been  scarcely  visible  throughout  the  years  1699,  1700,  1701,  at  thoso 
times  when  it  was  expected  to  be  most  conspicuous.  No.  59  Scuti  is 
sometimes  visible  to  the  naked  eye  at  its  minimum,  and  sometimes  not  so, 
and  its  maximum  is  also  very  irregular.    Figott's  variable  star  in  Corona 


472 


OUTLINES  OF  ASTRONOMY. 


is  stated  by  M.  Argelander  to  yary  for  the  most  part  bo  little  that  the 
UDaided  eye  can  hardly  decide  on  its  maxima  and  minima,  while  yet  after 
the  lapse  of  whole  years  of  these  slight  fluctuations,  they  suddenly  become 
so  great  that  the  star  completely  vanishes.  The  variations  of  a  Ononis, 
which  were  most  striking  and  unequivocal  in  the  years  1836 — 1840, 
within  the  years  since  elapsed  became  much  less  conspicuous.  They 
seem  now  (Jan.  1849)  to  have  recommenced. 

(827.)  These  irregularities  prepare  us  for  other  phsonomena  of  stellar 
variation,  which  have  hitherto  been  reduced  to  no  law  of  periodicity,  and 
must  be  looked  upon,  in  relation  to  our  ignorance  and  inexperience,  as 
altogether  casual;  or,  if  periodic,  of  periods  too  long  to  have  occurred 
more  than  once  within  the  limits  of  recorded  observation.  The  pbseno- 
mena  we  allude  to  are  those  of  Temporary  Stars,  which  have  appeared, 
from  time  to  time,  in  different  parts  of  the  heavens,  blazing  forth  with 
extraordinary  lustre ;  and  after  remaining  awhile  apparently  immoveable, 
have  died  away,  and  left  no  trace.  Such  is  the  star  which,  suddenly  ap- 
pearing some  time  about  the  year  125  b.  o.,  and  which  was  visible  in  the 
day-time,  is  said  to  have  attracted  the  attention  of  Hipparchus,  and  led 
him  to  draw  up  a  catalogue  of  stars,  the  earliest  on  record.  Such,  too, 
was  the  star  which  appeared,  A.D.  389,  near  a  AquilaD,  remaining  for 
three  weeks  as  bright  as  Venus,  and  disappearing  entirely.  In  the  years 
945, 1264,  and  1572,  brilliant  stars  appeared  in  the  region  of  the  heavens 
between  Cepheus  and  Cassiopeia;  and,  from  the  imperfect  account  vie 
have  of  the  places  of  the  two  earlier,  as  compared  with  that  of  the  last, 
which  was  well  determined,  as  well  as  fiom  the  tolerably  near  coincidence 
of  the  intervals  of  their  appearance,  we  may  suspect  them,  with  Good- 
ricke,  to  be  one  and  the  same  star,  with  a  period  of  312  or  perhaps  of 
156  jears.  The  appearance  of  the  star  of  1572  was  so  sudden,  that 
Tycho  Brahe,  a  celebrated  Danish  astronomer,  returning  one  evening  (the 
11th  of  November)  from  his  laboratory  to  his  dwelling-house,  was  sur- 
prised to  find  a  group  of  country  people  gazing  at  a  star,  which  he  was 
sure  did  not  exist  half  an  hour  before.  This  was  the  star  in  question. 
It  was  then  as  bright  as  Sirius,  and  continued  to  increase  till  it  surpassed 
Jupiter  when  brightest,  and  was  visible  at  mid-day.  It  began  to  diminish 
in  December  of  the  same  year,  and  in  March,  1574,  had  entirely  disap- 
peared. So,  also,  on  the  10th  of  October,  1604,  a  star  of  this  kind,  and 
not  less  brilliant,  burst  forth  in  the  constellation  of  Serpentarius,  which 
continued  visible  till  October,  1605. 

(828.)  Similar  phsenomena,  though  of  a  less  splendid  character,  have 
taken  place  more  recently,  as  in  the  case  of  the  star  of  the  third  magni- 
tude discovered  in  1670,  by  Anthelm,  in  the  head' of  the  Swan;  which, 


IRREaULAR  AND  TEMPOHART  STARS. 


478 


after  becoming  completely  invisible,  re-appcared,  and,  after  undergoing 
one  or  two  singular  fluctuations  of  light,  during  two  years,  at  lost  died 
away  entirely,  and  has  not  since  been  seen.  «.>:>;;> 

(829.)  On  the  night  of  the  28th  of  April,  1848,  Mr.  Hind  observed  a 
star  of  the  fifth  magnitude  or  5 '4  (very  conspicuous  to  the  naked  eye)  in 
a  part  of  the  constellation  Ophiuchus  (R.A.  16"  51"  l'-5.  N.P.D.  lOS** 
39'  14"),  where,  from  perfect  familiarity  with  that  region,  ho  was  certain 
that  up  to  the  fifth  of  that  month  no  star  so  bright  as  9*10  m.  previously 
existed.  Neither  has  any  record  been  discovered  of  a  star  being  there 
observed  at  any  previous  time.  From  the  time  of  its  discovery  it  con- 
tinued to  diminish,  without  any  alteration  of  place,  and  before  the 
advance  of  the  season  rendered  further  observation  impracticable,  was 
nearly  extinct.  Its  colour  was  ruddy,  and  was  thought  by  many 
observers  to  undergo  remarkable  changes,  an  e£fect  probably  of  its  low 
situation. 

(830.)  The  alterations  of  brightness  in  the  southern  star  ij  Argfis,  which 
have  been  recorded,  are  very  singular  and  surprising.  In  the  time  of  Halley 
(1677)  it  appeared  as  a  star  of  the  fourth  magnitude.  Lacaille,  in  1751, 
observed  it  of  the  second.  In  the  interval  from  1811tol815,it  was  again 
of  the  fourth ;  and  again  from  1822  to  1826  of  the  second.  On  the  1st 
of  February,  1827,  it  was  noticed  by  Mr.  Burchell  to  have  increased  to 
the  first  magnitude,  and  to  equal  a  Crucis.  Thence  again  it  receded  to 
the  second ;  and  so  continued  until  the  end  of  1887.  All  at  once  in  the 
beginning  of  1838  it  suddenly  increased  in  lustre  so  as  to  surpass  all  the 
stars  of  the  first  magnitude  except  Sirius,  Canopus,  and  a  Gentauri,  which 
last  star  it  nearly  equalled.  Thence  it  again  diminished,  but  this  time 
not  below  the  first  magnitude  until  April,  1843,  when  it  had  again 
increased  so  as  to  surpass  Canopus,  and  nearly  equc  "^  ^rius  in  splendour. 
"  A  strange  field  of  speculation,"  it  has  been  remarkt  J,  "  is  opened  by 
this  phaenomenon.  The  temporary  stars  heretofore  recorded  have  all 
become  totally  extinct.  Variable  stars,  so  far  as  they  have  been  carefully 
attended  to,  have  exhibited  periodical  alternations,  in  some  degree  at 
least  regular,  of  splendour  and  comparative  obscurity.  But  here  we  have 
a  star  fitfully  variable  to  an  astonishing  extent,  and  whose  fluctuations  are 
spread  over  centuries,  apparently  in  no  settled  period,  and  with  no  regu- 
larity of  progression.  What  origin  can  we  ascribe  to  these  sudden  flashes 
and  relapses?  What  conclusions  are  we  to  draw  as  to  the  comfort  or 
habitability  of  a  system  depending  for  its  supply  of  light  and  heat  on  so 
uncertain  a  source  ?"  Speculations  of  this  kind  can  hardly  be  termed 
visionary,  when  we  consider  that,  from  what  has  before  been  said,  we  are 
compelled  to  admit  a  commuui^^y  of  nature  between  the  fixed  stars  and  our 


1  ■ 

474 


OUTLINES  OF  ASTRONOMY. 


OWD  BUD ;  and  when  we  reflect  that  geology  testifies  to  the  fact  of  exten> 
sive  changes  having  taken  place  at  epochs  of  the  most  remote  antiquity  in 
the  climate  and  temperature  of  our  globe ;  changes  difficult  to  reconcile 
with  the  operation  of  secondary  causes,  such  as  a  different  distribution  ot 
sea  and  land,  but  which  would  find  an  easy  and  natural  explanation  in  a 
slow  variation  of  the  supply  of  light  and  heat  afiurded  primarily  by  the  sua 
itself. 

(881.)  The  Chinese  annals  of  Ma-touan-lin,'  in  which  stand  officiaVy 
recorded,  though  rudely,  remarkable  astronomical  pheenomena,  supply  a 
long  list  of  "strange  stars,"  among  which,  though  the  greater  part  are 
evidently  comets,  some  may  be  recognized  as  belonging  in  all  probability 
to  the  class  of  Temporary  Stars  as  above  characterized.  Such  is  that 
which  is  recorded  to  have  appeared  in  A.  D.  173,  between  a  and  /3  Cen- 
taurij  which  (no  doubt,  scintillating  from  its  low  situation)  exhibited 
« the  five  colours,"  and  remained  visible  from  December  in  that  year  till 
July  in  the  next.  And  another  which  these  annals  assign  to  A.  D.  1011, 
and  which  would  seem  to  be  identical  with  a  star  elsewhere  referred  to 
A.  D.  1012,  "  which  was  of  extraordinary  brilliancy,  and  remained  visible 
in  the  southern  part  of  the  heavens  during  three  months,'"  a  situation 
agreeing  with  the  Chinese  record,  which  places  it  low  in  Sagittarius. 
Among  several  less  unequivocal  is  one  referred  to  B.  0.  184,  in  Scorpio, 
which  may  possibly  have  been  Hipparchus's  star.  None  of  the  stars  o/ 
A.  D.  889,  945,  1264,  and  1572,  however,  are  noticed  in  these  records 
It  is  worthy  of  especial  notice,  that  all  the  stars  of  this  kind  on  record. 
of  which  the  places  are  distinctly  indicated,  have  occurred,  without  excep- 
tion, in  or  close  upon  the  borders  of  the  Milky  Way,  and  that  only  within 
the  following  semicircle,  the  preceding  having  offered  no  example  of  the 
kind. 

(832.)  On  a  careful  re-examination  of  the  heavens,  and  a  comparison 
of  catalogues,  many  stars  are  now  found  to  be  missing ;  and  although 
there  is  no  doubt  that  these  losses  have  arisen  in  the  great  majority  of 
instances  from  mistaken  entries,  and  in  some  from  planets  having  been 
mistaken  for  stars,  yet  in  some  it  is  equally  certain  that  there  is  no  mis- 
take in  the  observation  or  entry,  and  that  the  star  has  really  been  observed, 
and  as  really  has  disappeared  from  the  heavens.  The  whole  subject  of 
variable  stars  is  a  branch  of  practical  astronomy  which  has  been  too  little 
followed  up,  and  it  is  precisely  that  in  which  amateurs  of  the  science,  and 

'  Translated  by  M.  Edward  Biot,  Coiinoissance  des  Temps,  1846. 

'  Hind.  Notices  of  the  Astronomical  Society,  viii.  156,  citing  Hepidannus.  He  places 
the  Chinese  siar  of  173  b.c  between  a  and  &  CanU  Minorit,  but  M.  Biot  distinctly  says 
a  P  pied  oriental  du  Centaure, 


.  I 


DOUBLE  STARS. 


475 


especially  voyagera  at  sea,  provided  with  only  good  eyes,  or  moderate  !n- 
gtrumeots,  might  employ  their  time  to  ezcollent  advantage.  It  holds  out 
a  sure  promise  of  rich  discovery,  and  is  one  in  which  astronomers  in 
established  observatories  are  almost  of  necessity  precluded  from  taking  a 
part,  by  the  nature  of  the  observations  required.  Catalogues  of  the  com- 
parative brightness  of  the  stars  in  each  constellation  have  been  constructed 
by  Sir  Wm.  Herschel,  with  the  express  object  of  facilitating  these  re* 
searches,  and  the  reader  will  find  them,  and  a  full  account  of  his  method 
of  comparison,  in  the  Phil.  Trans.  1796,  and  subsequent  years.         •<  • 

(838.)  We  come  now  to  a  class  of  phaenomena  of  quite  a  different 
character,  and  which  give  us  a  real  and  positive  insight  into  the  nature 
of  at  least  some  among  the  stars,  and  enable  us  unhesitatingly  to  declare 
them  subject  to  the  same  dynamical  laws,  and  obedient  to  the  same  power 
of  gravitation  which  governs  our  own  system.  Many  of  the  stars,  when 
examined  with  telescopes,  are  found  to  be  double,  t.  e.  to  consist  of  two 
(in  some  cases  three  or  more)  individuals  placed  near  together.  This  might 
be  attributed  to  accidental  proximity,  did  it  occur  only  in  a  few  instances ; 
but  the  frequency  of  this  companionship,  the  extreme  closeness,  and,  in 
many  cases,  the  near  equality  of  the  stars  so  conjoined,  would  alone  lead 
to  a  strong  suspicion  of  a  more  near  and  intimate  relation  than  mere 
casual  juxtaposition.  The  bright  star.  Castor,  for  example,  when  much 
magnified,  is  found  to  consist  of  two  stars  of  nearly  the  third  magnitude, 
within  5"  of  each  other.  Stars  of  this  magnitude,  however,  are  not  so 
common  in  the  heavens  as  to  render  it  otherwise  than  excessively  impro- 
bable that,  if  scattered  at  random,  they  would  fall  so  near.  But  this  im- 
probability becomes  immensely  increased  by  a  consideration  of  the  fact,  that 
this  is  only  one  out  of  a  great  many  similar  instances.  Mitchell,  in  1767.. 
applying  the  rules  for  the  calculation  of  probabilities  to  the  case  of  tho 
six  brightest  stars  in  the  group  called  the  Pleiades,  found  the  odds  to  be 
500000  to  1  against  *ibeir  proximity  being  the  mere  result  of  a  random 
scattering  of  1500  stars  (which  he  supposed  to  be  the  total  number  of 
stars  of  that  magnitude  in  the  celestial  sphere')  over  the  heavens. 
Speculating  further  on  this,  as  an  indication  of  physical  connexion  rather 
than  fortuitous  assemblage,  he  was  led  to  surmise  the  possibility,  (since 
converted  into  a  certainty,  but  at  that  time,  antecedent  to  any  observation) 
of  the  existence  of  compound  stars  revolving  about  one  another,  or  rather 
about  their  common  centre  of  gravity.  M.  Struve,  pursuing  the  same 
train  of  thought  as  applied  specially  to  the  cases  of  double  and  triplo 

'  This  number  ia  considerably  too  small,  and  in  consequence,  Mitchell's  odds  in 
this  case  materially  overrated.  But  enough  will  remain,  if  this  be  rectified,  fully  to 
bear  out  his  argument.    Phil.  Trans,  vol.  57. 


Fll 


476 


OUTLINBS  OF  ASTRONOMT. 


/ 


oombinatioDfl  of  itara,  and  grounding  his  oomputationi  on  a  more  perfect 
enumeration  of  the  staru  visible  down  to  the  7  th  magnitude,  in  the  part 
of  the  heavena  visible  at  Dorpat,  calculates  that  the  odds  arc  0670  to  1 
against  any  two  stars,  from  the  let  to  the  7th  magnitude,  inclusive,  out 
of  the  whole  possible  number  of  binary  combinations  then  visible,  fulling, 
(if  fortuitously  scattered)  within  4"  of  each  other.  Now,  the  number 
of  instances  of  such  binary  combinations  actually  observed  at  the  date 
of  this  calculation  was  already  01,  and  many  more  have  since  been 
added  to  the  list.  Again,  he  calculates  that  the  odds  against  any 
such  stars  fortuitously  scattered,  falling  within  82"  of  a  third,  so  as  to 
constitute  a  triple  star,  is  not  less  than  173524  to  1.  Now,  four  such 
combinations  occur  in  the  heavens;  viz.  9  Orionis,  a  Ononis,  11  Monoce- 
rotis,  and  (  Cancri.  The  conclusion  of  a  physical  connexion  of  some  kind 
or  other  is  therefore  unavoidable. 

(834.)  Presumptive  evidence  of  another  kind  is  furnished  by  the  fol- 
lowing consideration.  Both  »  Centauri  and  61  Cygni  are  "  Double  Stars." 
Both  consist  of  two  individuals,  nearly  equal,  and  separated  from  each 
other  by  an  interval  of  about  a  quarter  of  a  minute.  In  the  case  of  61 
Cygni,  the  stars  exceeding  the  7th  magnitude,  there  is  already  a  prirnd 
facie  probability  of  0578  to  1  against  their  apparent  proximity.  The 
two  stars  of  a  Centauri  are  both  at  least  of  the  2d  magnitude,  of  which 
altogether  not  more  than  about  50  or  60  exist  in  the  whole  heavens, 
But,  waiving  this  consideration,  both  these  stars,  as  we  have  already  seen, 
have  a  proper  motion,  so  considerable  that,  supposing  the  condtitucnt  in- 
dividuals unconnected,  one  would  speedily  leave  the  other  behind.  Yet, 
at  the  earliest  dates  at  which  they  were  respectively  observed,  these  stars 
were  not  perceived  to  be  double,  and  it  is  only  to  the  employment  of  tele- 
scopes magnifying  at  least  8  or  10  times,  that  wo  owe  the  knowledge  we 
now  possess  of  their  being  so.  With  such  a  telescope,  Lacaille,  in  1751, 
was  barely  able  to  perceive  the  separation  of  the  two  constituents  of  a  Cen- 
tauri, whereas,  had  one  of  them  only  been  affected  with  the  observed 
proper  motion,  they  should  then  have  been  6'  asunder.  In  these  cases, 
then,  some  physical  connexion  may  be  regarded  as  proved  by  this  fact 
alone. 

(835.)  Sir  William  Herschel  has  enumerated  upwards  of  500  double 
stars,  of  which  the  individuals  are  less  than  32"  asunder.  M.  Struve, 
prosecuting  the  inquiry  with  instruments  more  conveniently  mounted  for 
the  purpose,  and  wrought  to  an  astonishing  pitch  of  optical  perfection, 
has  added  more  than  five  times  that  number.  And  other  observers  have 
extended  still  further  the  catalogue  of  "  Double  Stars,"  without  exhaust- 
ing the  fertility  of  the  heavens.     Among  these  are  a  great  many,  in 


DOUBLE  STARS. 


477 


which  the  distanoe  between  the  component  individuals  docs  not  exceed  a 
single  second.  They  are  divided  into  classes  by  M.  Stravo  (the  first  living 
authority  in  this  department  of  Astronomy/  a^ording  to  the  proximity 
of  their  component  individuals.  The  first  class  oomprises  those  only  in 
which  the  distance  does  not  exceed  1";  the  second  those  in  which  it  ex- 
ceeds 1",  but  falls  short  of  2";  the  8d  class  extends  from  2"  to  4"  dis- 
tance; the  4th  from  4"  to  8";  the  6th  from  8"  to  12";  the  6th  from  12" 
to  16";  the  7th  from  16"  to  24";  the  8th  from  24"  to  82".  Each  class 
he  again  subdivides  into  two  sub-classes  of  which  the  one  under  the  ap- 
pellation of  conspicuous  double  stars  (duplices  luoidse)  comprehends  those 
in  which  both  individuals  exceed  the  8^  magnitude,  that  is  to  say,  are 
teparately  bright  enough  to  be  easily  seen  in  any  moderately  good  tele- 
scope. All  others,  in  which  one  or  both  the  constituents  are  below  this 
limit  of  easy  visibility,  are  collected  into  another  sub-class,  which  he 
terms  residuary  (Duplices  reliquBe).  This  arrangement  is  so  far  conve- 
nient, that  after  a  little  practice  in  the  use  of  telescopes  as  applied  to 
such  objects,  it  is  easy  to  judge  what  optical  power  will  probal^';  suffice 
to  resolve  a  star  of  any  proposed  class  and  either  sub-class,  or  would  at 
least  be  so  if  the  second  or  residuary  sub-class  were  further  sub-divided 
by  placing  in  a  third  sub-class  "  delicate"  double  stars,  or  those  in  which 
the  companion  star  is  so  very  minute  as  to  require  a  high  degree  of  optical 
power  to  perceive  it,  of  which  instances  will  presently  be  given. 

(836.)  The  following  may  be  taken  as  specimens  of  each  class.  They 
are  all  taken  from  among  the  lucid,  or  conspicuous  stars,  and  to  such  of 
our  readers  as  may  be  in  possession  of  telescopes,  and  may  be  disposed  to 
try  them  on  such  objects,  will  afford  him  a  ready  test  of  their  degree  of 
efficiency. 


Class  I.,  0"  to  1". 


Y  Coronae  Bor. 

Y  Centauri. 

Y  Lupi. 

(  Arietig. 
{  Herculis. 


y  Circini. 
«  Cvgni. 
(  Chameleontia. 


1}  Coronae. 
1}  Herculis. 
X  CaBBiopeise. 
X  Ophiuchi. 
IT  Lupi. 


7  Ophiuchi. 
^  Draconis. 
^  UrssB  MtyoriB. 
X  Aquilffi. 
<■»  Leonis. 


Class  II.,  1"  to  2". 


(  Bootis. 
(  CassiopeigB. 
t  2  Cancri. 


(  UrsflB  M^joris. 
ir  Aquilae. 
9  Corons  Bor. 


Atlas  Pleiadum. 
4  Aquarii. 
42  ComtB. 
52  Arietis. 
66  Piaclum. 


2  CamelopardL 
32  Ononis. 
52  Ononis. 


■ll 

ll 

H|^g| 

Btfi'-V 

^wjj ff  ' 

■i,-'!^  \ 

'iii^^  1 

. 

'^i4  i 

' 

wcium. 


/3  HydrsB. 
y  Ceti. 
y  Loonis. 
Y  Coronae  Aus. 


Class  m.,  2"  to  4". 


y  Virginia. 
i  Serpentis. 
(  Bootis. 
c  Draconis. 
(  HydrsB. 


(  Aquarii. 
{  Orionis. 
I  Leonis. 
I  Trianguli. 
K  Lepons. 


fi  Draconia. 
]t  Canis. 
p  Herculis. 
or  Caaaiopeis. 
44  BootiB. 


liJf 


478 


OUTLINES   OF  ASTRONOMY. 


Class  IV.,  4"  to  8". 


/ 


a  Crucis. 
c  Herculifl. 
a  Geminorum. 
i  Geminorum. 
{  Coron»  Bor. 


/J  Orionis. 
y  Arietis. 
7  Delphini. 


•  Centauri. 
P  Ccphei. 
fi  Scorpii. 


a  Canum  Yen. 
«  Norms. 
{  Piscium. 


S  Herculis. 
17  LyrsB. 
(  Cancri. 


9  Phoenicia. 
«c  Cephei. 
X  Orionia. 
u  Cygni. 
i  Booiu. 


i  Cephei. 
»  Bootis. 
p  Capricorni. 
V  Argus, 
w  Auriga. 


U 


^  Eridani. 
70  Ophiuchi. 
12  Eridani. 
32  Eridani. 
95  Herculis. 


Class  v.,  8"  to  12". 

(  Antliae. 


n  uasaiopeis. 
0  Eridani. 

Class  VI.,  12"  to  16". 

y  Volantis. 

n  Lupi. 

i  Ursae  Major. 

Class  VII.,  16"  to  24". 

0  Serpentis. 
K  CoronsB  Aus. 
X  Tauri. 

Class  VIII.,  24"  to  32". 

K  Herculis. 
K  Cephei. 


1  Orionia. 
f  Eridani. 

2  Canum  Yen. 


•  K  Bootia. 
8  Monocerotis. 
61  CygnL 


24  Comee. 
41  Draconis. 
61  Ophiuchi. 


Y  Cygni. 
23  Ononis. 


'4/  Draconis. 

(837.)  Among  the  most  remarkable  triple,  quadruple,  or  multiple  stars 

(for  such  also  occur)  may  be  enumerated, 

a  Andromeds.  0  Orionis.  f  Scorpii. 

e  Lyrae.  fi  Lupi.  ^        11  Monocerotia. 

i  Cancri.  /i  Booiia.  '        12  Lyncia. 

Of  these  a  Andromedae,  ft  Bootis,  and  (i  Lupi,  appear  in  telescopes,  even 
of  considerable  optical  power,  only  as  ordinary  double  stars ;  and  it  is  only 
when  excellent  instruments  are  used  that  their  smaller  companions  are 
subdivided  and  found  to  be  in  fact,  extremely  close  double  stars.  <  Lyras 
offers  the  remarkable  combination  of  a  double-double  star.    Viewed  with 

Fij^  111. 


a  telescope  of  low  power  it  appears  as  a  coarse  and  easily  divided  double 
star,  but  on  increasing  the  magnifying  power,  each  individual  is  perceived 


OF  BINARY   STARS. 


479 


a  2  Cancri. 

a  Polaris. 

K  Circini. 

a  2  CapricomL 

fi  Aquanu 

K  Geminorum. 

a  Indi. 

y  Hydrse. 

H  Persei. 

u  LyrsB* 

UrssB  Majoris. 

7  Bootis. 

to  be  beautifully  and  closely  double,  the  one  pair  being  about  2^",  the 

other  about  3"  asunder.     Each  of  the  stars  ^  Cancri,  S  Soorpii,  11  Mono- 

cerotis,  and  12  Lyncis  consists  of  a  principal  star,  closely  double,  and  a 

smaller  and  more  distant  attendant,  while  9  Orionis  presents  the  phse- 

nomenon  of  four  brilliant  principal  stars,  of  the  respective  4th,  6th,  7th, 

■ri  Sth  magnitudes,  forming  a  trapezium,  the  longest  diagonal  of  which 

is  21".4,  and  accompanied  by  two  excessively  minute  and  very  close  com- 

panions  (as  in  the  annexed  figure),  to  perceive  both  which  is  one  of  the 

severest  tests  which  can  be  applied  to  a  telescope. 

(838.)  Of  the  "delicate"  sub-class  of  double  stars,  or  those  consisting 

of  very  large  and  conspicuous  principal  stars,  accompanied  by  very  minute 

companions,  the  following  specimens  may  suffice : 

^  Virginia. 
X  Eridani. 
16  Aurigffi. 
94  Ceti. 

(839.)  To  the  amateur  of  Astronomy  the  double  stars  offer  a  subject 
of  very  pleasing  interest,  as  tests  of  the  performance  of  his  telescopes, 
and  by  reason  of  the  finely  contrasted  colours  which  many  of  them  ex- 
hibit, of  which  more  hereafter.  But  it  is  the  high  degree  of  physical 
interest  which  attaches  to  them,  which  assigns  them  a  conspicuous  place 
in  modem  Astronomy,  and  justifies  the  minute  attention  and  unwearied 
diligence  bestowed  on  the  measurement  of  their  angles  of  position  and 
distances,  and  the  continual  enlargement  of  our  catalogues  of  them  by 
the  discovery  of  new  ones.  It  was,  as  we  have  seen,  under  an  impression 
that  such  combinations,  if  diligently  observed,  might  afford  a  measure 
of  parallax  through  the  periodical  variations  it  might  be  expected  to  pro- 
duce in  the  relative  situation  of  the  small  attendant  star,  that  Sir  W. 
Herschel  was  induced  (between  the  years  1779  and  1784)  to  form  the 
first  extensive  catalogues  of  them,  under  the  scrutiny  of  higher  magni- 
fying powers  than  had  ever  previously  been  applied  to  such  purposes.  In 
the  pursuit  of  this  object,  the  end  to  which  it  was  instituted  as  a  means 
was  necessarily  laid  aside  for  a  time,  until  the  accumulation  of  more 
abundant  materials  should  have  afforded  a  choice  of  stars  favourably  cir- 
cumstanced for  systematic  observation.  Epochal  measures  however,  of 
each  star,  were  secured,  and,  on  resuming  the  subject,  his  attention  was 
altogether  diverted  from  the  original  object  of  the  inquiry  by  phaenomena 
of  a  very  unexpected  character,  which  at  once  engrossed  his  whole  atten- 
tion. Instead  of  finding,  as  he  expected,  that  annual  fluctuation  to  and 
fro  of  one  star  of  a  double  star  with  respect  to  the  other, — that  alternate 
annual  increase  and  decrease  of  their  distance  and  angle  of  position,  which 
the  parallax  of  the  earth's  annual  motion  would  produce, — he  observed, 


?*.  ' 


..itj 


'<■•■. 


480 


OUTLINES  OF  ASTRONOMY. 


ia  many  instances,  a  regular  progressive  change;  in  some  cases  bearing 
chiefly  on  their  distance,  —  in  others  on  l^eir  position,  and  advancing 
steadily  in  one  direction,  so  as  clearly  to  indicate  either  a  real  motion  of 
the  stars  themselves,  or  a  general  rectilinear  motion  of  the  sun  and  whole 
;  solar  system,  producing  a  parallax  of  a  higher  order  than  would  arise  from 
the  earth's  orbitual  motion,  and  which  might  be  called  systematic 
parallax. 

(840.)  Supposing  the  two  stars,  and  also  the  sun,  in  motion  independ- 
ently of  each  other,  it  is  clear  that  for  the  interval  of  several  years,  these 
motions  must  be  regarded  as  rectilinear  and  uniform.  Hence,  a  very 
'  ..  iJight  acquaintance  with  gee  Jtietry  will  suffice  to  show  that  the  apparmt 
motion  of  one  star  of  a  double  star,  referred  to  the  other  as  a  centre,  and 
mapped  down,  as  it  were,  on  a  plane  in  which  that  other  shall  be  taken 
for  a  fixed  or  zero  point,  can  be  no  other  than  a  right  line.  This,  at 
least,  must  be  the  case  if  the  stars  be  independent  of  each  other ;  but  it 
ii^  1:  will  be  otherwise  if  they  have  a  physical  connexion,  such  as,  for  instance, 
'  real  proximity  and  mutual  gravitation  would  establish.  In  that  case,  they 
would  describe  orbits  round  each  other,  and  round  their  common  centre 
of  gravity ;  and  therefore  the  apparent  path  of  either,  referred  to  the  other 
as  fixed,  instead  of  being  a  portion  of  a  straight  line,  would  be  bent  into 
a  curve  concave  towards  that  other.  The  observed  motions,  however,  were 
so  slow,  that  many  years'  observation  was  required  to  ascertain  this  point; 
and  it  was  not,  therefore,  until  the  year  1803,  twenty-five  years  from  the 
commencement  of  the  inquiry,  that  any  thing  like  a  positive  conclusion 
could  be  come  to  respecting  the  rectilinear  or  orbitual  character  of  the 
observed  changes  of  position. 

(841.)  In  that,  and  the  subsequent  year,  it  was  distinctly  announced 
by  him,  in  two  papers,  which  will  be  found  in  the  Transactions  of  the 
Boyal  Society  for  those  years',  that  there  exist  sidereal  systems,  composed 
of  two  stars  revolving  about  each  other  in  regular  orbits,  and  constituting 
what  may  be  termed  binary/  stars,  to  distinguish  them  from  double  stars 
generally  so  called,  in  which  these  physically  connected  stars  are  con 
founded,  perhaps,  with  others  only  optically  double,  or  casually  juxta- 
.  posed  in  the  heavens  at  different  distances  from  the  eye ;  whereas  the  in* 
dividuals  of  a  binary  star  are,  of  course,  equidistant  from  the  eye,  or,  ai 
least,  cannot  differ  more  in  distance  than  the  semi-diameter  of  the  orbit 
they  describe  about  each  other,  which  is  quite  insignificant  compared  with 
the  immense  distance  between  them  and  the  earth.  Between  fifty  and 
sixty  insunces  of  changes,  to  a  greater  or  less  amount,  in  the  angles  of 

'  The  announcement  was  in  fact  made  in  1802,  but  unaccompanied  by  the  observa- 
Oona  eitablishing  the  fact. 


> 


OP  BINARY   STARS.   S  =; 


481 


positiou  .  (cuble  stars,  are  adduced  in  the  memoirs  above  mentioaed ; 
many  of  ^aich  are  too  decided,  and  too  regularly  progressive,  to  allow  of 
their  nature  being  misconceived.  In  particular,  among  the  more  con- 
spicuous stars, — Castor,  y  Virginis,  t  Ursae,  70  Ophiuchi,  <t  and  17  Coronae, 
I  Bootis,  ij  Cassiopeise,  y  Leonis,  ^  Herculis,  i  Cygni,  ft  Bootis,  >  4  and  < 
5  Lyrse,  x  Ophiuchi,  n  Draconis,  and  C  Aquarii,  are  enumerated  as  among 
the  most  remarkable  instances  of  the  observed  motion ;  and  to  some  of 
them  even  periodic  times  of  revolution  are  assigned  j  approximative  only, 
of  course,  and  rather  to  be  regarded  as  rough  guesses  than  as  results  of 
any  exact  calculation,  for  which  the  data  were  at  the  time  quite  inade- 
quate. For  instance,  the  revolution  of  Castor  is  set  down  at  334  years, 
that  of  y  Virginis  at  708,  and  that  of  y  Leonis  at  1200  years.  /> 

(842.)  Subsequent  observation  has  fully  confirmed  these  results.  Of 
all  the  stars  above  named,  there  is  not  one  which  is  not  found  to  be  fully 
entitled  to  be  regarded  as  binary ;  and,  in  fact,  this  list  comprises  nearly 
all  the  most  considerable  objects  of  that  description  which  have  yet  been 
detected,  though  (as  attention  has  been  closely  drawn  to  the  subject,  and 
observations  have  multiplied)  it  has,  of  late,  received  large  accessions. 
Upwards  of  a  hundred  double  stars,  certainly  known  to. possess  this  cha- 
racter, were  enumerated  by  M.  Madler  in  1841,'  and  more  are  emerging 
into  notice  with  every  fresh  mass  of  observations  which  come  before  the 
public.  They  require  excellent  telescopes  for  their  effective  observation, 
being  for  the  most  part  so  close  as  to  necessitate  the  use  of  very  high 
magnifiers  (such  as  would  be  considered  extremely  powerful  microscopes 
if  employed  to  examine  objects  within  our  reach),  to  perceive  an  interval 
between  the  individuals  which  compose  them. 

(843.)  It  may  easily  be  supposed,  that  phsenomena  of  this  kind  would 
not  pass  without  attempts  to  connect  them  with  dynamical  theories.  From 
their  first  discovery,  they  were  naturally  referred  to  the  agency  of  some 
power,  like  that  of  gravitation,  connecting  the  strrs  thus  demonstrated  to 
be  in  a  state  of  circulation  about  each  other;  tind  the  extension  of  the 
Newtonian  law  of  graxitation  to  these  remote  systems  was  a  step  so  ob- 
vious, and  so  well  warranted  by  our  experience  of  its  all-sufficient  agency 
in  our  own,  as  to  have  been  expressly  or  tacitly  made  by  every  one  who 
has  given  the  subject  any  share  of  his  attention.  We  owe,  however,  the 
first  distinct  system  of  calculation,  by  which  the  elliptic  elements  of  the 
orbit  of  a  binary  star  could  be  deduced  from  observations  of  its  angle  of 
position  and  distance  at  different  epochs,  to  M.  Savary,  who  showed*  that 
the  motions  of  one  of  the  most  remarkable  among  them  (|  Ursae)  were 

*  Dorpat  Observations,  vol.  ix.  1840  end  1841.  *  Connoiss.  des  Temps,  1830. 

81 


482 


OUTLINES   OF  ASTRONOMY. 


explicable,  within  the  limits  allowable  for  error  of  observation,  on  the 
supposition  of  an  elliptio  orbit  described  in  the  short  period  of  58^  years. 
A  different  process  of  computation  conducted  Professor  Encke'  to  an 
elliptic  orbit  for  70  Ophiuchi,  described  in  a  period  of  seventy-four  years. 
M.  Mttdler  has  especially  signalized  himself  in  this  line  of  inquiry  (sec 
note).  Several  orbits  have  also  been  calculated  by  Mr.  Hind  and  Cap- 
tain Smyth,  and  the  author  of  these  pages  has  himself  attempted  to  con- 
tribute his  mite  to  these  interesting  investigations.  The  following  may 
be  stated  as  the  chief  results  which  have  been  hitherto  obtained  in  this 
branch  of  astronomy:  — 

'  Berlin  Ephem.  1832. 

The  elementn  Nos.  1,  2,  3,  4  c,  5,  6  c,  7,  11  b,  12  a,  are  extracted  from  M.  Mad. 
ler's  synoptic  view  of  the  history  of  double  stars  in  vol.  ix.  of  the  Dorpat  Observations: 
4  a,  from  the  Connoiss.  des  Temps,  1830:  4  b,  6  b,  and  11  a,  from  vol.  v.  Trans 
Astron.  Soc.  Lend. :  6  a,  from  Berlin  Ephemeris,  1832 :  No.  8,  from  Trans.  Astron. 
Soc.  vol.  vi. :  No.  9,  11  c,  12  b,  and  13  from  Notices  of  the  Astronomical  Society, 
vol.  vii.  p.  22,  and  viii.  p.  159,  and  No.  10  from  the  author's  "  Results  of  Astronomicai 
Observations,  &c.  at  the  Cape  of  Good  Hope,"  p.  297.  The  2  prefixed  to  No.  7, 
denotes  the  number  of  the  star  in  M.  Struve's  Dorpat  Catalogue  (Catalogus  Novus 
Stellarum  Duplicium,  &.e,  Dorpat,  1827),  which  contains  the  places  for  1826  of  3112 
of  these  objects. 

The  "  position  of  the  node"  in  col.  4,  expresses  the  angle  of  position  (see  Art.  204) 
of  the  line  of  intersection  of  the  plane  of  the  orbit,  with  the  plane  of  the  heavens  on 
which  it  is  seen  projected.  The  "  inclination"  in  col.  6  is  the  inclination  of  these  two 
planes  to  one  another.  Col.  5  shows  the  an^le  actually  included  in  the  plane  of  the 
orbit,  between  the  line  of  nodes  (defined  as  above)  and  the  line  of  apsides.  The  ele- 
ments assigned  in  this  table  to  w  Leonis,  (  Bootis^  and  Castor  must  be  considered  as 
very  doubtful,  and  the  same  may  perhaps  be  said  of  those  ascribed  to  m  2  Bootis,  which 
rest  on  too  small  an  arc  of  the  orbit,  and  that  too  imperfectly  observed,  to  afford  a 
Mcure  basis  of  calculation. 


MX:< 


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ORBITS  OF  BINARY  8TARS. 


483 


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43 
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484 


OUTLINES  OF  ^SlL^OUOJi^T. 


/ 


(844.)  Of  the  stars  in  the  above  list,  that  which  has  been  most  assidu- 
ously watched,  and  has  offered  pbaenomenon  of  the  greatest  iuterest,  is 
y  Yirginis.  It  is  a  star  of  the  vulgar  8rd  magnitude  (3.08  =  Fbotom. 
8*494),  and  its  component  individuals  are  very  nearly  equal,  and  as  it 
would  seem  in  some  slight  degree  variable,  since,  according  to  the  obser- 
vations of  M.  Struve,  the  one  is  alternately  a  little  greater  and  a  little 
less  than  the  other,  and  occasionally  exactly  equal  to  it.  It  has  been 
known  to  consist  of  two  stars  since  the  beginning  of  the  eighteenth  cen- 
tury ;  the  distance  being  then  between  six  and  seven  seconds,  so  that  any 
tolerably  good  telescope  would  resolve  it.  When  observed  by  Herschel 
in  1780,  it  was  5"-66,  and  continued  to  decrease  gradually  and  regularly 
till  at  length,  in  1836,  the  two  stars  had  approached  so  closely  as  to  appear 
perfectly  round  and  single  under  the  highest  magnifying  power  which 
could  be  applied  to  most  excellent  instruments  —  the  great  refractor  at 
Pulkowa  alone,  with  a  magnifying  power  of  1000,  cootinuing  to  indicate, 
by  the  wedge-shaped  form  of  the  disc  of  the  star  its  Qomposite  nature. 
By  estimating  the  ratio  of  its  length  to  its  breadth  and  measuring  the 
former,  M.  Struve  concludes  that,  at  this  epoch  (1836*41),  the  distance 
of  the  two  stars,  centre  from  centre,  might  be  stated  at  0"*22.  From  that 
time  the  star  again  opened,  and  at  present  (1849)  the  individuals  are  more 
than  2"  asunder.  This  very  remarkable  diminution  and  subsequent  in- 
crease of  distance  has  been  accompanied  by  a  corresponding  and  equally 
remarkable  increase  and  subsequent  diminution  of  relative  angular  motion. 
Thus,  in  the  year  1783  the  apparent  angular  motion  hardly  amounted  to 
half  a  degree  per  annum,  while  in  1830  it  had  increased  to  5°,  in  1834 
to  20°,  in  1835  to  40°,  and  about  the  middle  of  1836  to  upwards  of  70° 
per  annum,  or  at  the  rate  of  a  degree  in  five  days.  This  is  in  entire  con- 
fermity  with  the  principles  of  dynamics,  which  establish  a  necessary  con- 
nexion between  the  angular  velocity  and  the  distance,  as  well  in  the 
apparent  as  in  the  real  orbit  of  one  body  revolving  about  another  under 
the  influence  of  mutual  attraction ;  the  former  varying  inversely  as  the 
square  of  the  latter,  whatever  be  the  curve  described  and  whatever  the 
law  of  the  attractive  force.  It  fortunately  happens  that  Bradley,  in  1718, 
had  noticed  and  recorded  in  the  margin  of  one  of  his  observation  books, 
the  apparent  direction  of  the  line  of  junction  of  the  two  stars,  as  seen  on 
the  meridian  in  his  transit  telescope,  viz.,  parallel  to  the  line  joining  two 
conspicuous  stars  a  and  j  of  the  same  constellation,  as  seen  by  the  naked 
eye.  This  note,  rescued  from  oblivion  by  the  late  Professor  Rigaud,  has 
proved  of  singular  service  in  the  verification  of  the  elements  above 
assigned  to  the  orbit,  which  represent  the  whole  series  of  recorded  obser- 
vations that  date  up  to  the  end  of  1846  (comprising  an  angular  movement 


Ij.ii 


ORBITS  OF  BINART  STARS. 


485 


of  nearly  nine-tenths  of  a  complete  circuit),  both  in  angle  and  distance, 
with  a  degree  of  exactness  fully  equal  to  that  of  observation  itself.  No 
doubt  can,  therefore,  remain  as  to  the  prevalence  in  this  remote  system  of 
the  Newtonian  law  of  gravitation. 

(845.)  The  observations  of  |  XJrsie  Majoris  are  equally  well  repre- 
sented by  M.  Madler's  elements  (4  c  of  our  table,)  thus  fully  justifying 
the  assumption  of  the  Newtonian  law  as  that  wh^ch  regulates  the  motions 
of  their  binary  systems.  And  even  should  it  be  the  case,  as  M.  Mddlcr 
appears  to  consider,  that  in  one  instance  at  least  (that  of  p  Ophiuchi,) 
deviations  from  elliptic  motion,  too  considerable  to  arise  from  mere  error 
of  observation,  exist  (a  position  we  are  by  no  means  prepared  to  grant,)' 
we  should  rather  be  disposed  to  look  for  the  cause  of  such  deviations  in 
perturbations  arising  (as  Bessel  has  suggested)  from  the  large  or  central 
star  itself  being  actually  a  close  and  hitherto  unrecognized  double  star 
than  in  any  defect  of  generality  in  the  Newtonian  law. 

(846.)  If  the  great  length  of  the  periods  of  some  of  these  bodies  be 
remarkable,  the  shortness  of  those  of  others  is  hardly  less  so.  f  Herculis 
has  already  completed  two  revolutions  since  the  epoch  of  its  first  discovery, 
exhibiting  in  its  course  the  extraordinary  spectacle  of  a  sidereal  occulta- 
tion,  the  small  star  having  twice  been  completely  hidden  behind  the  large 
one.  7  CoronoB,  ^  Cancri,  and  S  Ursae  have  each  performed  more  than 
one  entire  circuit,  and  70  Ophiuchi  and  y  Vir^nis  have  accomplished  by 
far  the  larger  portion  of  one  in  angular  motion.  If  any  doubt,  therefore, 
could  remain  as  to  the  reality  of  their  orbitual  motions,  or  any  idea  of 
explaining  them  by  mere  parallactic  changes,  or  by  any  other  hypothesis 
than  the  agency  of  centripetal  force,  these  facts  must  suffice  for  their  com  - 
plete  dissipation.  We  have  the  same  evidence,  indeed,  of  their  rotations 
about  each  other,  that  we  have  of  those  of  Uranus  and  Neptune  about  the 
sun ;  and  the  correspondence  between  their  calculated  and  observed  places 
in  such  very  elongated  ellipses,  must  be  admitted  to  carry  with  it  proof  of 
the  prevalence  of  the  Newtonian  law  of  gravity  in  their  systems,  of  the 
very  same  nature  and  cogency  as  that  of  the  calculated  and  observed  places 
of  comets  round  the  central  body  of  our  own. 

(847.)  But  it  is  not  with  the  revolutions  of  bodies  of  a  planetary  or 

'  p  Ophiuchi  belot.gs  to  the  class  of  very  unequal  double  stars,  the  magnitudes  of  the 
individuals  being  4  and  7.  Such  stars  present  difficulties  in  the  exact  measurement  of 
their  angles  of  positior.  which  even  yet  continue  to  embarrass  the  observer,  though, 
owing  to  later  improvements  in  the  art  of  executing  such  measurements,  their  influ- 
ence is  confined  within  much  narrower  limits  than  in  the  earlier  history  of  the  subject. 
In  simply  placing  a  fine  single  wire  parallel  to  the  line  of  junction  of  two  such  stars  it 
is  easily  possible  to  commit  an  error  of  3°  or  4".  By  placing  them  between  two  parallel 
(hick  wires  such  errors  are  in  great  measure  obviated. 


[ 


486 


OUTLINES  OF  ASTRONOMT. 


cometary  nature  round  a  solar  centre  that  we  are  now  concerned ;  it  u 
with  that  of  sun  round  sun  —  each,  perhaps,  at  least  in  some  binary  sys- 
tems where  the  individuals  are  very  remote  and  their  period  of  revolution 
very  long,  accompanied  with  its  train  of  planets  and  their  satellites,  closely 
shrouded  from  our  view  by  the  splendour  of  their  respective  suns,  and 
crowded  into  a  space  bearing  hardly  a  greater  proportion  to  the  enormous 
interval  which  separates  them,  than  the  distances  of  the  satellites  of  our 
planets  from  their  primaries  bear  to  their  distances  from  the  sun  itself. 
A  less  distinctly  characterized  subordination  would  be  incompatible  with 
the  stability  of  their  systems,  and  with  the  planetary  nature  of  their  orbits. 
Unless  closely  nestled  under  the  protecting  wing  of  their  immediate  supe- 
rior, the  sweep  of  their  other  sun  in  its  perihelion  passage  round  their 
own  might  carry  them  off,  or  whirl  them  into  orbits  utterly  incompatible 
with  the  conditions  necessary  for  the  existence  of  their  inhabitants.  It 
must  be  confessed,  that  we  have  here  a  strangely  wide  and  novel  field 
for  speculative  excursions,  and  one  which  it  is  not  easy  to  avoid  luxu- 
riating in. 

(848.)  The  discovery  of  the  parallaxes  of  a  Centauri  and  61  Cygni, 
both  which  are  above  enumerated  among  the  "  conspicuous"  double  st^rs 
of  the  6th  class  (a  distinction  fully  merited  in  the  case  of  the  former  by 
the  brilliancy  of  both  its  constituents),  enables  us  to  speak  with  an  ap- 
proach to  certainty  as  to  the  absolute  dimensions  of  both  their  orbits,  and 
thence  to  form  a  probable  opinion  as  to  the  general  scale  on  which  these 
astonishing  systems  are  constructed.  The  distance  of  the  two  stars  of  61 
Cygni  subtends  at  the  earth  an  angle  which,  since  the  earliest  micro- 
metrical  measures  in  1781,  has  varied  hardly  half  a  second  from  a  mean 
value  15"'5.  On  the  other  hand,  the  angle  of  position  has  altered  since 
the  same  epoch  by  nearly  50°,  so  that  it  would  appear  probable  that  the 
true  form  of  the  orbit  is  not  far  from  circular,  its  situation  at  right  angles 
to  the  visual  line,  and  its  periodic  time  probably  not  short  of  500  years. 
Now,  as  the  ascertained  parallax  of  this  star  is  0"-348,  which  is,  there- 
fore, the  angle  the  radius  of  the  earth's  orbit  would  subtend  if  equally 
remote,  it  follows  that  the  mean  distance  between  the  stars  is  to  that 
radius,  as  15"-5  :  0"-348,  or  as  44-54  : 1.  The  orbit  described  by  these 
two  stars  about  each  other  undoubtedly,  therefore,  greatly  exceeds  in 
dimensions  that  described  by  Neptune  about  the  sun.  Moreover,  suppo- 
sing the  period  to  be  five  centuries  (and  the  distance  being  actually  on  the 
increase,  it  can  hardly  be  less)  the  general  propositions  laid  down  by 
Newton',  taken  in  conjunction  with  Kepler's  third  law,  enable  us  to  calcu- 
late the  sum  of  the  masses  of  the  two  stars,  which,  on  these  data  we  find 

* Principia,  L  i.    Prop.  57,  58.  59.'  .  ,,,    ;   , ,  _.> 


COLOUB'iSD  DOUBLE  STARS. 


487 


to  l>e  0-858,  the  mass  of  our  sun  being  1.     The  sun,  therefore,  is  neither 
vastly  greater  nor  vastly  less  than  the  stars  composing  61  Cygni. 

(849.)  The  data  in  the  case  of  a  Centauri  are  more  uncertain.  Since 
the  year  1822,  the  distance  has  been  steadily  and  pretty  rapidly  decreasing 
at  the  rate  of  about  half  a  second  per  annum,  and  that  with  very  little 
change  in  the  angle  of  position.  Hence,  it  follows  evidently  that  the 
plane  of  its  orbit  passes  nearly  through  the  earth,  and  (the  distance  about 
the  middle  of  1834  having  been  17V')  it  is  very  probable  that  either  an 
occultution,  like  that  observed  in  ^  Herculis,  on  a  close  appulse  of  the 
two  stars,  will  take  place  about  the  year  1867.  As  the  observations  wo 
possess  afford  no  sufficient  grounds  for  a  satisfactory  calculation  of  elliptic 
elements'  we  must  be  content  to  assume  what,  at  all  events,  they  fully 
justify,  viz.,  that  the  major  semiaxis  must  exceed  12",  and  is  very  pro- 
bably considerably  greater.  Now  this  with  a  parallax  of  0"-913  would 
give  for  the  real  value  of  the  semiaxis  13 -15  radii  of  the  earth's  orbit,  as 
a  minimum.  The  real  dimensions  of  their  ellipse,  therefore,  cannot  be  so 
small  as  the  orbit  of  Saturn ;  in  all  probability  exceeds  that  of  Uranus ; 
and  may  possibly  be  much  greater  than  either. 

(850.)  The  parallel  between  these  two  double  stars  is  a  remarkable  one. 
Owing  no  doubt  to  their  comparative  proximity  to  our  system,  their  appa- 
rent propr  motions  are  both  unusually  great,  and  for  the  same  reason 
probably  rather  than  owing  to  unusually  large  dimensions,  their  orbits 
appear  to  us  under  what,  for  binary  double  stars,  we  must  call  unusually 
large  angles.  Each  consists,  moreover,  of  stars,  not  very  unequal  in 
brightness,  and  in  each  both  the  stars  are  of  a  high  yellow  approaching 
to  orange  colour,  the  smaller  individual,  in  each  case,  being  also  of  a 
deeper  tint.  Whatever  the  diversity,  therefore,  which  may  obtain  among 
other  sidereal  objects,  these  would  appear  to  belong  to  the  same  family  or 

genus.*        "''. ^. -:.;,,, -v:-.v^  ■.-:■-..  ;-,:;v  h-.,^-'^  ::^ 

(851.)  Many  of  the  double  stars  exhibit  the  curious  and  beautiful 
phaenomenon  of  contrasted  or  complementary  colours.'    In  such  instances, 


'  Elements  have  been  recently  computed  by  Captain  Jacob,  for  which  see  the  table, 

p.  483.  ^      ■.-_,•.--  ■         _  ..^  .. 

^  Similar  combinations  are  very  numerous.  Many  remarkable  instances  occur  among 
the  double  stars  catalogued  by  the  author  in  the  2nd,  3rd,  4th,  6th  and  9th  volumes  of 
Trans.  Roy.  Ast.  Soc.  and  in  the  volume  of  Southern  observations  already  cited.  See 
Nos.  121,  375, 1066, 1907,  2030,  2146,  2244,  2772,  3863,  3396,  3998,  4000,  4055,  4196, 
4210,  4615,  4649,  4765,  5003,  5012,  of  these  catalogues.  The  fine  binary  star,  B.  A.  C. 
No.  4923,  has  its  constituents  13"  apart,  the  one  6m.  yellow,  the  other,  7m.  orange. 
» "        ■         other  suns,  perhaps. 

With  their  attendant  moons,  thou  wilt  descry. 
Communicating  male  and  female  light,  .  .  <.  .       i    .> 


488 


0UTLINX8  OF  A8TR01T0MT. 


the  larger  star  is  usually  of  a  ruddy  or  orange  hue,  while  the  smaller  one 
appears  blue  or  green,  probably  in  virtue  of  that  general  law  of  optics, 
which  provides,  that  when  the  retina  is  under  the  influence  of  excitement 
by  any  bright,  coloured  light ;  feebler  lights,  which  seen  alone  would  pro- 
duce no  sensation  but  of  whiteness,  shall  for  the  time  appear  coloured  with 
the  tint  complementary  to  that  of  the  brighter.  Thus  a  yellow  colour 
predominating  in  the  light  of  the  brighter  star,  that  of  the  less  bright  one 
in  the  same  field  of  view  will  appear  blue;  while,  if  the  tint  of  the 
brighter  star  verges  to  crimson,  that  of  the  other  will  exhibit  a  tendency  to 
green — or  even  appear  as  a  vivid  green,  under  favourable  circumstances. 
The  former  contrast  is  beautifully  exhibited  by  »  Oancri — the  latter  by  y 
Andromeilec',  both  fine  double  stars.  If,  however,  the  coloured  star  bo 
much  the  less  bright  of  tho  two,  it  will  not  materially  affect  the  other. 
Thus,  for  instance,  tj  Cassiopeise  exhibits  the  beautiful  combination  of  a 
large  white  star,  and  a  small  one  of  a  rich  ruddy  purple.  It  is  by  no 
means,  however,  intended  to  say,  that  in  all  such  cases  one  of  the  colours 
is  a  mere  effect  of  contrast,  and  it  may  be  easier  suggested  in  words,  than 
conceived  in  imagination,  what  variety  of  illumination  two  suns  —  a  red 
and  a  green,  or  a  yellow  and  a  blue  one — must  afford  a  planet  circuktiDg 
about  either ;  and  what  charming  contrasts  and  "  grateful  vicissitudes," 
—  a  red  and  a  green  day,  for  instance,  alternating  with  a  white  one  and 
with  darkness, — might  urise  from  tho  presence  or  absence  of  one  or  other, 
or  both,  above  the  horizon.  Insulated  stars  of  a  red  colour,  almost  as 
deep,  as  that  of  blood,'  occur  in  many  parts  of  the  heavens,  but  no  green 
or  blue  star  (of  any  decided  hue)  has,  we  believe,  ever  been  noticed  un- 
associated  with  a  companion  brighter  than  itself.  Many  of  the  red  stars 
are  variable.     ■  7   v         ,-■■■/- 

(Which  two  gretit  sexes  animate  the  world,) 
Stored  in  each  orb,  perhaps,  with  some  that  live." 

Paradue  Last,  viii.  148. 
'  The  small  star  of  y  Andromedae  is  close  double.    Both  its  individuals  are  green :  1 
similar  comhinatior.,  with  even  more  decided  colours,  is  presented  by  the  double  star, 
h.  881. 

'  The  following  are  the  R.  ascensions  and  N.  P.  distances  for  1830,  of  some  of  the 


nost  remar 

kaule  ot  tnese  e 

anguine  or  rul 

jy  stars : — 

-  ,     \,-  - 

R.A. 

N.  P.  D.     > 

R.A. 

N.P.D. 

R.A. 

N.  P.  D. 

h.  m.  s. 

0    /     t> 

h.  m.  s. 

0    /     n 

h.  m.  8. 

0    /    It 

4  40  53 

4  51  51 

5  38  29 
9  27  56 

61  46  21 
102    2    4 
136  32  15 
152    2  48 

9  48  31 
10  52  10 
12  37  31 
16  29  44 

130  47  12 
107  24  40 
148  45  47 
122    2    0 

20  7    8 

21  37  18 
21  37  20 

111  50  11 
31  59  47 
52  54  47 

Of  these  No.  5  (in  order  of  right  ascension)  is  in  the  same  field  of  view  with  a  Hydra, 
and  No.  9  with  fi  Crucis.    No.  2  (in  the  same  order)  is  variable. 


PROPER  MOTIONS  OF  THE   STARS. 


//489 


(852.)  Another  very  interesting  subject  of  inquiry,  in  the  physical 
history  of  the  stars,  is  their  proper  motion.     It  was  first  noticed  by 
Halley,  that  three  principal  stars,  Sirius,  Areturus,  and  Aldebaran,  arc 
placed  by  Ptolemy,  on  the  strength  of  observations  made  by  Hipparchus, 
180  years  B.C.,  in  ktitudes  respectively  20',  22',  and  88'  more  northerlj/ 
than  he  actually  found  them  in  1717.'     Making  due  allovrauco  for  the 
diminution  of  obliquity  of  the  ecliptic  in  the  interval  (1847  years)  they 
ougLi;  to  have  stood,  if  really  fixed,  respectively  10',  14',  and  O'  moro 
southerly.     As  the  circumstances  of  the  statement  exclude  the  supposi* 
tion  of  error  of  transcription  in  the  MSS.,  wo  are  necessitated  to  admit  a 
southward  motion  in  latitude  in  these  stars  to  the  very  considerable  extent, 
respectively,  of  87',  42',  and  88',  and  this  is  corroborated  by  an  observa- 
tion of  Aldebaran  at  Athens,  in  the  year  A.  d.  509,  which  star,  on  tho 
11th  of  March  in  that  year,  was  seen  immediately  after  its  emergence 
from  occultation  by  the  moon,  in  such  a  position  as  it  could  not  have  had 
if  the  occultation  were  not  nearly  central.     Now,  from  the  knowledge  we 
have  of  tho  lunar  motions,  this  could  not  have  been  the  case  had  Alde- 
baran at  that  time  so  much  southern  latitude  as  at  present.     A  priori,  it 
might  be  expected  that  apparent  motions  of  some  kind  or  other  should 
be  detected  among  so  great  a  multitude  of  individuals  scattered  through 
space,  and  with  nothing  to  keep  them  fixed.     Their  mutual  attractions 
even,  however  inconceivably  enfeebled  by  distance,  and  counteracted  by 
opposing  attractions  from  opposite  quarters,  must  in  the  lapse  of  count- 
less ages  produce  some  movements — some  change  of  internal  arrangement 
— resulting  from  the  difierence  of  the  opposing  actions.     And  it  is  a  fact, 
that  such  apparent  motions  are  really  proved  to  exist  by  the  exact  obser- 
vations of  modern  astronomy.     Thus,  as  we  have  seen,  the  two  stars  of 
61  Cygni  have  remained  constantly  at  the  same,  or  very  nearly  the  same, 
distance,  of  15",  for  at  least  fifty  years  past,  although  they  have  shifted 
their  local  situation  in  the  heavens,  in  this  interval  of  time,  through  no 
less  than  4'  23",  the  anniuil  proper  motion  of  each  star  being  5"-C ;  by 
which  quantity  (exceeding  a  third  of  their  interval)  this  system  is  every 
year  carried  bodily  along  in  some  unknown  path,  by  a  motion  which,  for 
many  centuries,  must  be  regarded  as  uniform  and  rectilinear.     Among 
stars  not  double,  and  no  way  differing  from  the  rest  in  any  other  obvious 
particular, «  Indi'  and  ji*  Cassiopeiaa  are  to  be  remarked  as  having  the 
greatest  proper  motions  of  any  yet  ascertained,  amounting  respectively  to 
7" -74  and  3" -74  of  annual  displacement.     And  a  great  many  others  hav(t 


»  Phil.  Trans.  1717,  vol.  mx.  fo.  736. 
•  D' Arrest.  Astr.  Nachr.,  No.  618. 


400 


I 


OUTLINES  or  A8TR0N01IT. 


been  observed  to  be  thus  constantly  carried  awaj  from  their  places  by 
smaller,  but  not  less  unequivocal  motions.'  '^ 

(858.)  Motions  which  require  whole  centuries  to  accumulate  boforo 
they  produce  changes  of  arrangement,  such  nn  the  naked  o,Vj  can  detect, 
though  quite  sufficient  to  destroy  that  idea  of  roatho*-  "lioal  hxity  which 
precludes  speculation,  are  yet  too  trifling,  as  far  as  pruotical  applications 
go,  to  induce  a  change  of  language,  and  lead  us  to  speak  of  the  st^rs  m 
common  parlance  as  otherwise  than  fixed.  Small  as  they  arc,  however, 
astronomers,  once  assured  of  their  reality,  have  not  been  wanting  in  aU 
tempts  to  explain  and  reduce  them  to  general  laws.  No  one,  who  reflects 
with  due  attention  on  the  subject,  will  be  inclined  to  deny  the  high  probn 
bility,  nay  certainty,  that  the  sun  as  well  as  the  stars  must  have  a  pn  .  i 
motion  in  some  direction;  and  the  inevitable  consequence  of  suc^  a  .u* 
tion,  if  uupartioipated  by  the  rest,  must  be  a  slow  averag  app■><v^f^  teu 
dency  of  all  the  stars  to  the  vanishing  point  of  lines  par.ui(.i  lo  that 
direction,  and  to  the  region  which  he  is  leaving,  how  /er  greatly  indi- 
vidual  stars  might  differ  from  such  average  by  reason  of  their  own  pecu- 
liar proper  motion.  This  is  the  necessary  effect  of  perspective ;  and  it  is 
certain  that  it  must  be  detected  by  observation,  if  we  knew  accurately  tho 
apparent  proper  motions  of  all  the  stars,  and  if  we  were  sure  that  they 
were  independent,  i.  e.  that  the  whole  firmament,  or  at  least  all  that  part 
which  we  see  in  our  own  neighbourhood,  were  not  drifting  along  together, 
by  a  general  set  as  it  wore,  in  one  direction,  the  result  of  unknown  pro- 
cesses and  slow  intoraal  changes  going  on  in  the  sidereal  stratum  to 
which  our  system  belong'^,  as  we  see  motes  sailing  in  a  current  of  air, 
and  keeping  nearly  the  same  relative  situation  with  respect  to  one  another. 

(854.)  It  wa£i  on  this  assumption,  tacitly  made  indeed,  but  nccc£>sarily 
implied  in  every  step  of  his  reasoning,  that  Sir  William  Ilerschcl,  in 
1783,  on  a  consideration  of  the  apparent  proper  motions  of  such  stars  as 
could  at  that  period  be  considered  as  tolerably  (though  still  imperfectly) 
ascertained,  arrived  at  the  conclusion  that  a  relative  motion  of  the  sun, 
among  the  fixed  stars  in  the  direction  of  a  point  or  parallactic  apex,  situ- 
ated near  h  Herculis,  that  is  to  say,  in  R.  A.  iV  l:M'»-^260°  34',  N.  P.  D. 
<53°  43'  (1790),  would  ;i«count  for  the  chi.  I"  vWr  ,  ^pparen>  .uotions, 
leaving,  however,  some  still  outstanding  anu  uuL  explicable  by  this  cause; 
and  in  the  same  year  Prevost,  taking  nearly  the  same  view  of  the  subject, 
arrived  at  a  conclusion  as  to  the  solar  apex  (or  point  of  the  sphere  towards 
which  the  sun  relatively  advances),  agreeing  nearly  in  polar  distance  with 

'  Tb-  reader  nay  consult  "a  list  of  314  stars  having,  or  supposed  to  have,  a  proper 
mnf-  i\0'  not  le-K  .han  about  0"'5  of  a  great  circle"  (per  annum)  by  the  late  F.  Baily, 
Key*    Trans,  A  «*.  See,  v.  p.  158.  ;  \ 


MOTION  OF  THB  SUN  IN  SPACE. 


491 


the  foregoing,  bat  diifori>i?  from  it  about  27**  in  right  asoension.  Since 
that  time  methods  of  calculuti  n  have  been  improved  and  concinnated, 
our  knowledge  of  tbu  pro;  ^>-  motiuu.^  if  the  stars  has  been  rendered  more 
precise,  and  a  greater  nuni^xT  of  cases  o  such  motions  have  l)een  re- 
corded. The  subject  <  >s  been  u..su<iiod  hy  several  eminent  a:  onoroers 
and  mathematicians:  viz.  Int,  by  M.  Argeluu  lor,  who,  from  the  "onside- 
ration  of  the  proper  motions  of  21  stars  ezcoediug  i  per  annum  arc, 
has  placed  the  solar  apex  in  R.  A.  266°  2^',  N.  P.  D.  51"  28' ;  from  >  yae 
of  50  stars  between  0"5  and  T'-O,  in  26i>  10'  61°  26';  and  from  those 
of  319  stars  having  motions  between  0"'l  n  )  0"-5  j/or  annum,  in  261® 
"!!'  59"  2' :  2ndly,  by  M.  Luhndahl,  whose  Jculations,  founded  on  the 
proper  motions  of  147  stars,  give  262°  68',  75°  84' :  and  8rdly,  by  M. 
Otto  Struvo,  whoso  result  261°  22',  62°  24',  emer  -qs  fro,  i  a  very  elabo- 
rate discussion  of  the  proper  motions  of  892  stars.  Al:  these  places  aro 
for  A.  D.  1790. 

(855.)  The  most  probable  mean  of  the  results  obtai  aed  by  these  threo 
astronomers,  is  (for  the  same  epoch)  R.  A. =259°  9',  S.  P.  D.  65°  28'. 
Their  researches,  however,  extending  only  to  stars  vi  ole  in  European 
observatories,  it  became  a  point  of  high  interest  to  ascertain  how  far  the 
stars  of  the  southern  hemisphere  not  so  visible,  treated  ii  dependently  on 
the  same  system  of  procedure,  would  corroborate  or  contr-  <^rt  their  con- 
clusion. The  observations  of  Lacaille,  at  the  Cape  of  C  ^  Hope,  in 
1751  and  1762,  compaud  with  those  of  Mr.  Johnson  at  >  t.  Helena,  in 
1829-33,  and  of  Henderson  at  the  Cape  in  1880  and  1831,  lave  afforded 
the  means  of  deciding  this  question.  The  task  has  very  t  ^cently  been 
executed  in  a  masterly  manner  by  Mr.  Galloway,  in  a  paper  tublished  in 
the  Philosophical  Transactions  for  1847  (to  which  we  may  al  o  refer  the 
reader  for  a  more  particular  account  of  the  history  of  the  subject  than  our 
limits  allow  us  to  give.)  On  comparing  the  records,  Mr.  Galloway  finds 
eighty-one  southern  stars  not  employed  in  the  previous  investigations  above 
referred  to,  whose  proper  motions  in  the  intervals  elapsed  appear  consider- 
able enough  to  assure  us  that  they  have  not  originated  in  error  of  the 
earlit  •  observations.  Subjecting  these  to  the  same  process  of  computation 
he  concludes  for  the  place  of  the  solar  apex,  for  1790,  as  follows :  viz. 
R.  A.  260°  1',  N.  P.  D.  55°  87',  a  result  so  nearly  identical  with  that 
afforded  by  the  northern  hemisphere,  as  to  afford  a  full  conviction  of  its 
v'Mir  upproach  to  truth,  and  what  may  fairly  be  considered  a  demonstration 
ot  the  physical  cause  assigned. 

(850 )  Of  the  mathematical  conduct  of  this  inquiry  the  nature  of  this 
^ork  precludes  our  giving  any  account ;  but  as  the  philosophical  principle 
^  wbicU  it  is  based  has  been  misconceived,  it  is  necessary  to  say  a  few 


i 


492 


OUTLINES  OF  ASTRONOMY. 


words  ia  explanatioa  of  it.     Almost  all  the  greatest  discoveries  in  astron- 
omy bave  resulted  from  the  consideration  of  what  we  have  elsewhere 
termed  residual  phenomena',  of  a  quantitative  or  numerical  kind, 
that  is  to  say,  of  such  portions  of  the  numerical  or  quantitative  results  of 
observation  as  remain  outstanding  and  unaccounted  for  after  subductiog 
and  allowing  for  all  that  would  result  from  the  strict  application  of  known 
principles.     It  was  thus  that  the  grand  discovery  of  the  precession  of  the 
equinoxes  resulted  as  a  residual  pbasnomenon,  from  the  imperfect  explana- 
tion of  the  return  of  the  seasons  by  the  return  of  the  sun  to  the  same 
apparent  place  among  the  fixed  stars.    Thus,  also,  aberration  and  nutation 
resulted  as  residual  phasnomena  from  that  portion  of  the  changes  of  the 
apparent  places  of  the  fixed  stars  which  was  left  unaccounted  for  by  pre- 
cession.    And  thus  again  the  apparent  proper  motions  of  the  stars  are 
the  observed  residues  of  their  apparent  movements  outstanding  and  unac- 
counted for  by  strict  calculation  of  the  effects  of  precession,  nutation,  and 
aberration.     The  nearest  approach  which  human  theories  can  make  to 
perfection  is  to  diminish  this  residue,  this  caput  mortuum  of  observation, 
as  it  may  be  considered,  as  much  as  practicable,  and,  if  possible,  to  reduce 
it  to  nothing,  either  by  showing  that  something  has  been  neglected  in  our 
estimation  of  known  causes,  or  by  reasoning  upon  it  as  a  new  fact,  and  on 
the  principle  of  the  inductive  philosophy  ascending  from  the  effect  to  its 
cause  or  causes.     On  the  suggestion  of  any  new  cause  hitherto  unresorted 
to  for  its  explanation,  our  first  object  must  of  course  be  to  decide  whether 
such  a  caiise  would  produce  such  a  result  in  kind :  the  next,  to  assign  to 
it  such  an  intensity  as  shall  account  for  the  greatest  possible  amount  of  the 
residual  matter  in  hand.    The  proper  motion  of  the  sun  being  suggested 
as  such  a  cause,  we  have  two  things  disposable — its  direction  and  velocity, 
both  which  it  is  evident,  if  they  ever  became  known  to  us  at  all,  can  only 
be  so  by  the  consideration  of  the  very  phaenomena  in  question.    Our 
object,  of  course,  is  to  account,  if  possible,  for  the  wJiole  of  the  observed 
proper  motions  by  the  proper  assumption  of  these  elements.     If  this  he 
inapracticable,  what  remains  iinaccounted  for  is  a  residue  of  a  more  recon- 
dite kind,  but  which,  so  long  as  it  is  unaccounted  for,  we  must  regard  as 
purely  casual,  seeing  that,  for  anything  we  can  perceive  to  the  contrary, 
it  might  with  equal  probability  be  one  way  as  the  other.     The  theory  of 
chances,  therefore,  necessitates  (as  it  does  in  all  such  cases)  the  application 
ot  a  general  mathematical  process,  known  as  "  the  method  of  least  squares," 
which  leads,  as  a  matter  of  strict  geometrical  conclusion,  to  the  values  of 
the  elements  sought,  whichf  under  all  the  circumstances,  are  the  viost 
probable. 

■    >  Discourse  on  the  Study  of  Natural  Philosophy.    Cab.  Cyclopadia,  No  14. 


PROPER  MOTION   OP  THE   SUN. 


493 


(857.)  This  is  the  process  resorted  to  by  all  the  geometers  we  have 
enumerated  in  the  foregoing  articles  (arts.  854,  855).     It  gives  not  only 
the  direction  in  space,  but  also  the  velocity  of  the  solar  motion,  estimated 
on  a  scale  conformable  to  that  in  which  the  velocity  of  the  sidereal  motions 
to  be  explained  are  given ;  i.  e.  in  seconds  of  arc  as  subtended  at  the 
average  distance  of  the  stars  concerned,  by  its  annual  motion  in  space. 
But  here  a  consideration  occurs  which  tends  materially  to  complicate  the 
problem,  and  to  introduce  into  its  solution  an  element  depending  on  sup- 
positions more  or  less  arbitrary.     The  distance  of  the  stars  being,  except 
in  two  or  three  instances,  unknown,  we  are  compelled  either  to  restrict  our 
inquiry  to  these,  which  are  too  few  to  ground  any  result  on,  or  to  make 
some  supposition  as  to  the  relative  distances  of  the  several  stars  employed. 
In  this  we  have  nothing  but  general  probability  to  guide  us,  and  two 
courses  only  present  themselves,  either,  1st,  To  class  the  distances  of  the 
stars  according  to  their  magnitudes,  or  apparent  brightnesses,  and  to 
institute  separate  and  independent  calculations  for  each  class,  including 
stars  assumed  to  be  equidistant,  or  nearly  so :  or,  2dly,  To  class  them 
according  to  the  observed  amount  of  their  apparent  proper  motions,  on 
the  presumption  that  those  which  appear  to  move  fastest  are  really  nearest 
to  us.    ^he  former  is  the  course  pursued  by  M.  Otto  Struve,  the  latter 
by  M.  Argelander.     With  regard  to  this  latter  principle  of  classification, 
however,  two  considerations  interfere  with  its  applicability,  viz.  1st  that 
we  see  the  real  motion  of  the  stars  foreshortened  by  the  effect  of  perspec- 
tive; and  2dly,  that  that  portion  of  the  total  apparent  proper  motion 
which  arises  from  the  real  motion  of  the  sun  depends,  not  simply  on  the 
absolute  distance  of  the  star  from  the  sun,  but  also  on  its  angular  apparent 
distance  from  the  solar  apex,  being,  cseteris  paribus,  as  the  sine  of  that 
angle.     To  execute  such  a  classification  correctly,  therefore,  we  ought  to 
know  both  these  particulars  for  each  star.     The  first  is  evidently  out  of 
our  reach.     We  are  therefore,  for  that  very  reason,  compelled  to  regard  it 
as  casual,  and  to  assume  that  on  the  average  of  a  great  number  of  stars  it 
would  be  uninfluential  on  the  result.     But  the  second  cannot  be  so  sum- 
marily disposed  of.     By  the  aid  of  an  approximate  knowledge  of  the  solar 
apex,  it  is  true,  approximate  values  may  be  found  of  the  simply  apparent 
portions  of  the  proper  motions,  supposing  all  the  stars  equidistant,  and 
these  being  subducted  from  the  total  observed  motions,  the  residues  might 
afford  ground  for  the  classification  in  question.'     This,  however,  would  be 

'  M.  Argelander's  classes,  however,  are  constructed  without  reference  to  this  con- 
sideration, on  the  sole  basis  6f  the  total  apparent  amount  proper  motion,  and  are,  there- 
fore, pro  lanto,  questionable.  It  is  the  more  satisfactory  theii  to  find  so  considerabia 
an  agreement  among  his  partial  results  as  actually  obtains. 


r-.  H 


'<■'-•  ill 


lii 


V\ 


494 


OUTLINES  OF    /.STRONOMY. 


a  long,  and  to  a  certain  extent  precarious  system  of  procedure.  On  the 
other  hand,  the  classification  by  apparent  brightness  is  open  to  no  such 
difficulties,  since  we  are  fully  justified  in  assuming  that,  on  a  general 
average,  the  brighter  stars  are  the  nearer,  and  that  the  exceptions  to  this 
rule  are  casual  in  that  sense  of  the  word  which  it  always  bears  iu  such 
inquiries,  expressing  solely  our  ignorance  of  any  ground  for  assuming  a 
bias  one  way  or  other  on  either  side  of  a  determinate  numerical  rule.  In 
Mr.  Galloway's  discussion  of  the  southern  stars  the  consideration  of  dis- 
tance is  waived  altogether,  which  is  equivalent  to  an  admission  of  complete 
ignorance  on  this  point,  as  well  as  respecting  the  real  directions  and 
velocities  of  the  individual  motions. 

(858.)  The  velocity  of  the  solar  motion  which  results  from  M.  Otto 
Struve's  calculations  is  such  as  would  carry  it  over  an  angular  subtense 
of  0"-3392  if  seen  at  right  angles  from  the  average  distance  of  a  star  of 
the  first  magnitude.  If  we  take,  with  M.  Struve,  senior,  the  parallax  of 
such  a  star  as  probably  equal  to  0"-209,'  we  shall  at  once  be  enabled  to 
compare  this  annual  motion  with  the  radius  of  the  earth's  orbit,  the  result 
being  1-623  of  such  units.  The  sun  then  advances  through  space  (rela- 
tively, at  least,  among  the  stars,)  carrying  with  it  the  whole  planetary 
and  cometary  system  with  a  velocity  of  1-623  radii  of  the  eartH's  orbit, 
or  154,185,000  miles  per  annum,  or  422,000  miles  (that  is  to  say, 
nearly  its  own  semi-diameter)  per  diem :  in  other  words,  with  a  velocity 
a  very  little  greater  than  one-fourth  of  the  earth's  annual  motion  in  its 
orbit. 

(859.)  Another  generation  of  astronomers,  perhaps  many,  must  pass 
away  before  we  are  in  a  condition  to  decide  from  a  more  precise  and 
extensive  knowledge  of  the  proper  motions  of  the  stars  than  we  at  present 
possess,  how  far  the  direction  and  velocity  above  assigned  to  the  solar 
motion  deviates  from  exactness,  whether  it  continue  uniform,  and  whether 
it  show  any  sign  of  deflection  from  rectilinearity ;  so  as  to  hold  out  a 
prospect  of  one  day  being  enabled  to  trace  out  an  arc  of  the  solar  orbit, 
and  to  indicate  the  direction  in  which  the  preponderant  gravitation  of  the 
sidereal  firmanent  is  urging  the  central  body  of  our  system.  An  analogy 
for  such  deviation  from  uniformity  would  seem  to  present  itself  in  the 
alleged  existence  of  a  similar  deviation  in  the  proper  motions  of  Sirius 
and  Procyon,  both  which  stars  are  considered  to  have  varied  sensibly  in 
this  respect  within  the  limits  of  authentic  and  dependable  observation. 
Such,  indeed,  would  appear  to  be  the  amount  of  evidence  for  this  as  a 
matter  of /act  as  to  have  given  rise  to  a  speculation  on  the  probable  circu- 
lation of  these  stars  round  opaque  (and  therefore  invisible)  bodies  at  no 
*  Etudes  d'Aetronomie  Stellaire,  p.  107. 


SPECULATIONS   ON   A  CENTRAL  SUN. 


495 


■  ■!:>  ■.»'  : 


y'i-i; 


great  distances  from  them  respectively,  in  the  manner  of  binary  stars : 
[and  it  has  been  recently  shown  by  M.  Peters  (Ast.  Nachr.  748,)  that, 
in  the  case  of  Sirius,  such  a  circulation,  performed  in  a  period  of  50093 
years  in  an  ellipse  whose  excentricity  is  0.7994,  the  perihelion  passago 
taking  place  at  the  epoch  A.  D.  1701-431,  would  reconcile  in  a  remark- 
able manner  the  observed  anomalies,  and  reduce  the  residual  motion  to 
uniformity.] 

(860.)  The  whole  of  the  reasoning  upon  which  the  determination  of 
the  solar  motion  in  space  rests,  is  based  upon  the  entire  exclusion  of  any 
law  either  derived  from  observation  or  assumed  in  theory,  affecting  the 
amount  and  direction  of  the  real  motions  both  of  the  sun  and  stars.  It 
supposes  an  absolute  non-recognition,  in  those  motions,  of  any  general 
directive  cause,  such  as,  for  example,  a  common  circulation  of  all  about  a 
common  centre.  Any  such  limitation  introduced  into  the  conditions  of 
the  problem  of  the  solar  motion  would  alter  in  toto  both  its  nature  and 
the  form  of  its  solution.  Suppose,  for  instance,  that,  conformably  to  the 
speculations  of  several  astronomers,  the  whole  system  of  the  Milky  Way, 
including  our  sun,  and  the  stars,  our  more  immediate  neighbours,  which 
constitute  our  sidereal  firmament,  should  have  a  general  movemeut  of  rota- 
tion in  the  plane  of  the  galactic  circle  (any  other  would  be  exceedingly 
improbable,  indeed  hardly  reconcilable  with  dynamical  principles,)  being 
held  together  in  opposition  to  the  centrifugal  force  thus  generated  by  the 
mutual  gravitation  of  its  constituent  stars.  Except  we  at  the  same  time 
admitted  that  the  scale  on  which  this  movement  proceeds  is  so  enormous 
that  all  the  stars  whose  proper  motions  we  include  in  our  calculations  go 
together  in  a  body,  so  far  as  that  movement  is  concerned  (as  forming  too 
small  an  integrant  portion  of  the  whole  to  differ  sensibly  in  their  relation 
to  its  central  point;)  we  stand  precluded  from  drawing  any  conclusion 
whatever,  not  only  respecting  the  absolute  motion  of  the  sun,  bui  respect 
ing  even  its  relative  movement  among  those  stars,  until  we  have  established 
some  law,  or  at  all  events  framed  some  hypothesis  having  the  provisional 
force  of  a  law,  connecting  the  whole,  or  a  part  of  the  motion  of  each  indi» 
vidual  with  its  situation  in  space. 

(861.)  Speculations  of  this  kind  have  not  been  wanting  in  astronomy, 
and  recently  an  attempt  has  been  made  by  M.  Miidler  to  assign  the  local 
centre  in  space,  round  which  the  sun  and  stars  revolve,  which  he  places 
in  the  group  of  the  Pleiades,  a  situation  in  itself  improbable,  lying  as  it 
does  no  less  than  26°  out  of  the  plane  of  the  galactic  circle,  out  of  which 
plane  it  is  almost  inconceivable  that  any  general  circulation  can  take  place. 
In  the  present  defective  state  of  our  knowledge  respecting  the  proper 
motion  of  the  smaller  stars,  especially  in  right  ascension^  (an  element  for 


f>    ■ 


L-^ 


496 


OUTLINES  OF  ASTRONOMY. 


/' 


the  moat  part  far  less  exactly  ascertainable  than  the  polar  distance,  or  at 
least  which  has  been  hitherto  far  less  accurately  ascertained,)  we  cannot 
but  regard  all  attempts  of  the  kind  as  to  a  certain  extent  premature, 
though  by  no  means  to  be  discouraged  as  forerunners  of  something  more 
decisive.  The  question,  as  a  matter  of  fact,  whether  a  rotation  of  the 
galaxy  in  its  own  plane  exist  or  not,  might  be  at  once  resolved  by  the 
assiduous  observation,  both  in  K.  A.  and  polar  distance,  of  a  considerable 
number  of  stars  of  the  Milky  Way,  judiciously  selected  for  the  purpose, 
and  including  all  magnitudes,  down  to  the  smallest  distinctly  identifiable, 
and  capable  of  being  observed  with  normal  accuracy :  and  we  would  re- 
commend the  inquiry  to  the  special  attention  of  the  directors  of  permanent 
observatories,  provided  with  adequate  instrumental  means,  in  both  hemi- 
spheres. Thirty  or  forty  years  of  observation,  perseveringly  directed  to 
the  object  in  view,  could  not  fail  to  settle  the  question.' 

(862.)  The  solar  motion  through  space,  if  real  and  not  simply  relative, 
must  give  rise  to  uranographical  corrections  analogous  to  parallax  and 
aberration.  The  solar  ur  systematic  parallax  is  no  other  than  that  part 
of  the  proper  motion  of  each  star  which  is  simply  apparent,  arising  from 
the  sun's  motion,  and  until  the  distances  of  the  stars  be  known,  must 
remain  inextricably  mixed  up  with  the  other  or  real  portion.  The  syste- 
matic aberration,  amounting  at  its  maximum  (for  stars  90°  from  the  solar 
apex)  to  about  5",  displaces  all  the  stars  in  gi'eat  circles  diverging  from 
that  apex  through  angles  proportional  to  the  sines  of  their  respective  dis- 
tances from  it.  This  displacement,  however,  is  permanent,  and  therefore 
uncognizable  by  any  phaenoraenon,  so  long  as  the  solar  motion  remains 
invariable ;  but  should  it,  in  the  course  of  ages,  alter  its  direction  and 
velocity,  both  the  direction  and  amount  of  the  displacement  in  question 
would  alter  with  it.  The  change,  however,  would  become  mixed  up  with 
other  changes  in  the  apparent  proper  motions  of  the  stars,  and  it  would 
seem  hopeless  to  attempt  disentangling  them. 

(863.)  A  singular,  and  at  first  sight  paradoxical  effect  of  the  progres- 
sive movement  of  light,  combined  with  +he  proper  motion  of  the  stars,  is 
that  it  alters  the  apparent  periodic  time  in  which  the  individuals  of  a 
liinary  star  circulate  about  each  other.'  To  make  this  apparent,  suppose 
them  to  circulate  round  each  other  in  a  plane  perpendicular  to  the  visual 

*  An  examination  of  the  proper  motions  of  the  stars  of  the  B.  Assoc.  Catal.  in  the 
portion  of  the  Milky  Way  nearest  either  pole  (where  the  motion  should  be  almost 
wholly  in  R  A)  indicates  no  distinct  symptom  of  such  a  rotation.  If  the  question  be 
taken  up  fundamentally,  it  will  involve  a  redetermination  from  the  recorded  proper 
motions,  both  of  the  precession  of  the  equinoxes  Cjid  the  change  of  obliquity  of  the 
ecliptic. 

•  Astronomische  Nachrichten,  No.  520. 


SPECULATIONS  ON  A  CENTRAL  SUN. 


.497 


ray  in  a  period  of  10,000  days.  Then  if  both  the  sun  and  the  centre  of 
gravity  of  the  binary  system  remained  fixed  in  space,  the  relative  apparent 
situation  of  the  stars  would  be  exactly  restored  to  its  former  state  after 
the  lapse  of  this  interval,  and  if  the  angle  of  position  were  0°  at  first, 
after  10,000  days  it  would  again  be  so.  But  now  suppose  thai  the  centre 
of  gravity  of  the  star  were  in  the  act  of  receding  in  a  direct  line  from 
the  sun,  with  a  velocity  of  one-tenth  part  of  the  radius  of  the  earth's 
orbit  per  diem.  Then  at  the  expiration  of  10,000  days  it  would  be  more 
remote  from  us  by  1000  such  radii,  a  space  which  light  would  require  57 
days  to  traverse.  Although  really,  therefore,  the  stars  would  have  arrived 
at  the  position  0"  at  the  exact  expiration  of  10,000  days,  it  would  require 
57  days  more  for  the  notice  of  that  fact  to  reach  our  system.  In  other 
words,  the  period  would  appear  to  us  to  be  10,057  days,  since  we  could 
only  conclude  the  period  to  be  completed,  when  to  us,  as  observers,  the 
original  angle  of  position  was  again  restored.  A  contrary  motion  would 
produce  a  contrary  efieot. 


1 

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498 


OUTLINES  OF  ASTRONOMY. 


w 


CHAPTER  XVII. 
OP    CLUSTERS    OP    STARS    AND    NEBULA. 

OF  CLUSTERING  GROUPS  OF  STARS. — GLOBULAR  CLUSTERS.  —  THEIR 
STABILITY  DYNAMICALLY  POSSIBLE.  —  LIST  OF  THE  MOST  REMARK- 
ABLE. —  CLASSIFICATION  OF  NEBULAE  AND  CLUSTERS.  —  THEIR 
DISTRIBUTION  OVER  THE  HEAVENS.  —  IRREGULAR  CLUSTERS. — 
RESOLV ABILITY  OF  NEBULiE.  —  THEORY  OF  THE  FORMATION  OP 
Oi.USTERS     BY    NEBULOUS     SUBSIDENCE.  —  OF    ELLIPTIC    NEBULA. 

—  THAT    OF  ANDROMEDA. — ANNULAR  AND    PLANETARY   NEBULiE. 

—  DOUBLE  NEBULiE.  —  NEBULOUS  STARS. — CONNEXION  OF  NEBULAE 
WITH  DOUBLE  STARS.  —  INSULATED  NEBULA,  OF  FOBUfS  NOT 
WHOLLY  IRREGULAR.  —  OF  AMORPHOUS  NEBUL.X.  —  THEIR  LAW  OF 
DISTRIBUTION  MARKS  THEM  AS  OUTLIERS  OF  THE  GALAXY.— 
NEBULA,  AND  NEBULOUS  GROUP  OF  ORION.  —  OP  ARGO.  —  OP  SA- 
GITTARIUS.—  OF  CYGNUS.  —  THE  MAGELLANIC  CLOUDS.  —  SINGULAR 
NEBULA  IN  THE  GREATER  OF  THEM. — THE  ZODIACAL  LtGHT.— 
SHOOTING   STARS. 

(864.)  When  we  cast  our  eyes  over  the  concave  of  the  heavons  iu  a  clear 
night,  we  do  not  fail  to  observe  that  here  and  there  are  gruups  of  stars 
which  seem  to  be  compressed  together  in  a  more  condensed  manner  than 
in  the  neighbouring  parts,  formJag  bright  patches  and  clusters,  which 
attract  attention,  as  if  they  were  there  brought  together  by  some  general 
cause  other  than  casual  distribution.  There  is  a  group,  called  the  Pleiades, 
in  which  six  or  seven  stars  may  be  noticed,  if  the  eye  be  directed  full 
upon  it ;  and  many  more  if  the  eye  he  turned  careless/)/  aside,  while  the 
attention  is  kept  directed '  upon  the  group.  Telescopes  show  fifty  or  sixty 
large  stars  thus  crowded  together  in  a  very  moderate  space,  comparatively 

'  It  is  a  very  remarkable  fact,  that  the  centre  of  the  visual  area  is  far  less  sensible 
to  feeble  impressions  of  light,  than  the  exterior  portions  of  the  retina.  Few  persons 
are  aware  of  the  extent  to  which  this  comparative  insensibility  extends,  previous  to 
trial  To  estimate  it,  let  the  reader  look  alternately  full  at  a  star  of  the  tnih  niagiii- 
Hide,  and  beside  it ;  or  choose  two  equally  bright,  and  about  3°  or  4°  apart,  and  look 
full  at  one  of  them,  the  probability  is  he  will  see  only  the  other.  The  fact  accounis  for 
the  multitude  of  stars  with  which  we  are  impressed  by  a  general  view  of  the  heavens ; 
their  paucity  when  we  come  to  count  them. 


insula 
Bereu 
whole 
(8( 
less  d 
model 
tirely 
spot, 
into  ii 
stars ; 


CLUSTERS  OF  STARS  AND  NEBULJ!. 


499 


insulated  from  the  rest  of  the  heavenu.  The  constellation  called  Coma 
Berenices  is  another  such  group,  more  diffused,  and  consisting  on  the 
whole  of  larger  stars. 

(865.)  In  the  constellation  Cancer,  there  is  a  somewhat  similar,  hut 
less  definite,  luminous  spot,  called  Prsesepe,  or  the  hee-hive,  which  a  very 
moderate  telescope,  —  an  ordinary  night-glass  for  instance, — resolves  en- 
tirely into  stars.     In  the  sword-handle  of  Perseus,  also,  is  another  such 
spot,  crowded  with  stars,  which  requires  rather  a  better  telescope  to  resolve 
into  individuals,  separated  from  each  other.     These  are  called  clusters  of 
stars ;  and,  whatever  be  their  rature,  it  is  certain  that  other  laws  of  ag- 
gregation subsist  in  these  spots,  than  those  which  have  determined  the 
scattering  of  stars  over  the  general  surface  of  the  sky.    This  conclusion 
is  still  more  strongly  pressed  upon  us,  when  we  come  to  bring  very 
powerful  telescopes  to  bear  on  these  and  similar  spots.     There  are  a 
great  number  of  objects  which  have  been  mistaken  for  comets,  and,  in 
fact,  have  very  much  the  appearance  of  comets  without  tails :  small  round, 
or  oval  nebulous  specks,  which  telescopes  of  moderate  power  only  show 
as  such.     Messier  has  given,  in  the  Connois.  des  Temps  for  1784,  a  list 
of  the  places  of  103  objects  of  this  sort ;  which  all  those  who  search  for 
comets  ought  to  be  familiar  with,  to  avoid  being  misled  by  their  similarity 
of  appearance.     That  they  are  not,  however,  comets,  their  fixity  suflS- 
ciently  proves ;  and  when  we  come  to  examine  them  with  instruments  of 
great  power,  —  such  as  reflectors  of  eighteen  inches,  two  feet,  or  more  in 
aperture, — any  such  idea  is  completely  destroyed.    They  are  then,  for  the 
most  part,  perceived  to  consist  entirely  of  stars  crowded  together  so  as  to 
occupy  almost  a  definite  outline,  and  to  run  up  to  a  blaze  of  light  in  the 
centre,  where  their  condensation  is  usually  the  greatest.     (See  Jiff.  1, 
pi.  II.,  which  represents  (somewhat  rudely)  the  thirteenth  nebula  of 
Messier's  list  (described  by  him  as  ndbulmise  sans  dtaiks),  as  seen  in  a 
reflector  of  18  inches  aperture  and  20  feet  focixl  length.)     Many  of  them, 
indeed,  are  of  an  exactly  round  figure,  and  convey  the  complete  idea  of  a 
globular  space  filled  full  of  stars,  insulated  in  the  heavens,  and  constitutr 
ing  in  itself  a  family  or  society  apart  from  the  rest,  and  subject  only  to 
its  own  internal  laws.     It  would  be  a  vain  task  to  attempt  to  count  the 
stars  in  one  of  these  globular  clusters.     They  are  not  to  be  reckoned  by 
hundreds;  and  on  a  rough  calculatior,  grounded  on  the  apparent  intervala 
between  them  at  the  borders,  and  the  angular  diameter  of  the  whole  group, 
it  would  appear  that  many  clusters  of  this  description  must  contain  at 
least  five  thousand  stars,  compacted  and  wedged  together  in  a  round  space, 
whose  angular  diameter  does  not  exceed  eight  or  ten  minutes ;  that  is  to 
say,  in  an  area  not  more  than  a  tenth  part  of  that  covered  by  the  moon. 


I  \r 


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500 


OUTLINES   OP  ASTRONOMY. 


(866.)  Perhaps  it  may  be  thought  to  savour  of  the  gigantesque  to  look 
upon  the  individuals  of  such  a  group  as  suns  like  our  own^  and  their  mu- 
tual distances  as  equal  to  those  which  separate  our  sun  from  the  nearest 
fixed  star :  yet,  when  we  consider  that  their  united  lustre  affects  the  eye 
with  a  less  impression  of  light  than  a  star  of  the  fourth  magnitude,  (for 
the  largest  of  these  clusters  is  barely  visible  to  the  nake''  eye,)  the  idea 
we  are  thus  compelled  to  form  of  their  distance  from  us  may  prepare  us 
for  almost  any  estimate  of  their  dimensions.  At  all  events,  we  can 
hardly  look  upon  a  group  thus  insulated,  thus  in  seipso  totus,  teres,  afque 
rofundus,  as  not  forming  a  system  of  a  peculiar  and  definite  character. 
Thr^ir  round  figure  clearly  indicates  the  existence  of  some  general  bond 
of  union  in  the  nature  of  an  attractive  force ;  and,  in  many  of  lucm,  there 
is  an  evident  acceleration  in  the  rate  of  condensation  as  we  approach  the 
centre,  which  is  not  referable  to  a  merely  uniform  distribution  of  equidis- 
tant stars  through  a  globular  space,  but  marks  an  intrinsic  density  in  their 
state  of  aggregation,  greater  in  the  centre  than  at  the  surface  of  the  mass. 
It  is  difficult  to  form  any  conception  of  the  dynamical  state  of  such  a 
system.  On  the  one  hand,  without  a  rotatory  motion  and  a  centrifugal 
force,  it  is  hardly  possible  not  to  regard  them  as  in  a  state  of  progressive 
collapse.  On  the  other,  granting  such  a  motion  and  such  a  foroe,  we  find 
it  no  less  difficult  to  reconcile  the  apparent  sphericity  of  their  form  with 
a  rotation  of  the  whole  system  round  any  single  axis,  without  which  in- 
ternal collisions  might  at  first  sight  appear  to  be  inevitable.  If  we  sup- 
pose a  globular  space  filled  with  equal  stars,  uniformly  dispersed  tlirough 
it,  and  very  numerous,  each  of  them  attracting  every  other  with  a  force 
inversely  as  the  square  of  the  distance,  the  resultant  force  by  which  any 
one  of  them  (those  at  the  surface  alone  excepted)  will  bo  urged,  in  virtue 
of  their  joint  attractions,  will  be  directed  towards  the  common  centre  of 
the  sphere,  and  will  be  directly  as  the  distance  therefrom.  This  follows 
from  what  Newton  has  proved  of  the  internal  attraction  of  a  homogeneous 
sphere.  (&C9  also  note  on  Art.  735.)  Now,  under  such  a  law  of  force, 
each  particulai  star  would  describe  a  perfect  ellipse  about  the  common 
centre  of  gravity  as  its  centre,  and  that,  in  whatever  plane  and  whatever 
direction  it  might  i*evolve.  The  condition,  therefore,  of  a  rotation  of  the 
cluster.  ::ti  a  mass,  about  a  single  axis  would  be  unnecessary.  Each 
ellipse,  whatever  might  be  the  proportion  of  its  axis,  or  the  inclination 
of  its  plane  to  the  others,  would  be  invariable  in  every  particular,  and  all 
would  be  described  in  one  common  period,  so  that  at  the  end  of  every 
such  period,  or  annus  magnua  of  the  system,  every  star  of  the  cluster 
(except  the  superficial  ones)  would  be  exactly  re-established  in  its  original 
position,  thence  to  set  out  afresh,  and  run  the  same  unvarying  round  for 


an  m( 
be  so 
each 


// 


CLASSIFICATION  OF  NEBULJE. 


501 


an  indcGaite  succession  of  ages.  Supposing  their  motions,  therefore,  to 
be  so  adjusted  at  any  one  moment  as  that  the  orbits  shv.  '  not  intersect 
each  other,  and  so  that  the  magnitude  of  each  star,  and  tne  sphere  of  its 
more  intense  attraction,  should  bear  but  a  small  proportion  to  the  distance 
separating  the  individuals,  such  a  system,  it  is  obvious,  might  subsist,  and 
realize,  in  great  measure,  that  abstract  and  ideal  harmony,  Tvhich  NewtoD| 
in  the  89  th  Proposition  of  the  First  Book  of  the  Principia,  has  shown 
to  characterize  a  law  of  force  directly  as  the  distance.' 

(867.)  The  following  are  the  places,  for  1830,  of  the  principal  of  these 
remarkable  objects,  as  specimens  of  their  class :  — 


R.  A. 

N.  P.  D. 

R.  A. 

N.  P.  D. 

R.  A. 

N.  P.  D. 

h.  m.  8. 

0    / 

h.  m.  8. 

o    / 

h.  m.  B. 

0    t 

0  16  25 

163   2 

15   9  56 

87  16 

17  26  51 

143  34 

9   8  33 

154  10 

15  34  56 

127  13 

17  28  42 

93   8 

12  47  41 

159  67 

16   6  5f 

112  33 

11  26   4 

114   2 

13   4  30 

70  65 

16  23   2 

102  40 

18  65  49 

150  14 

13  16  38 

136  35 

16  35  37 

53  13 

21  21  43 

78  34 

13  U  10 

60  46 

16  50  24 

119  61 

21  24  40 

91  34 

Of  these,  by  far  the  most  conspicuous  and  remarTiable  is  «  Centauri  the 
fifth  of  the  list  in  order  of  Eight  Ascension.  It  is  visible  to  the  naked 
eye  as  a  dim  round  cometic  object  about  equal  to  a  star  4'5  m.j  though 
probably  ii"  concentered  in  a  single  point,  the  impression  on  the  eye  would 
be  much  greater.  Viewed  in  a  powerful  telescope  it  appears  as  a  globe 
of  fully  20'  in  diameter,  very  gradually  increasing  in  brightness  to  the 
centre,  and  composed  of  innumerable  stars  of  the  13th  and  15th  magni- 
tudes (the  former  probably  being  two  or  more  of  the  latter  closely  juxta- 
posed). The  11th  in  order  of  the  list  (R.  A.  16"  35")  is  also  visible  to 
the  naked  eye  in  veiy  fine  nights,  between  jj  and  f  Herculis,  and  is  a 
superb  object  in  a  large  telescope.  Both  were  discovered  by  Halley,  the 
former  in  1677,  and  the  latter  in  1714. 

(808.)  It  is  to  Sir  William  Herschel  that  we  owe  the  most  complete 
analysis  of  the  great  variety  of  those  objects  which  are  generally  classed 
under  the  common  head  of  Nebi^lae,  but  which  have  been  separated  by 
him  into  —  1st.  Clusters  of  stars,  in  which  the  stars  are  clearly  distin- 
guishable; and  these,  again,  into  globular  and  irregular  clusters;  2d. 
Kesolvable  nebulae,  or  such  as  excite  a  suspicion  that  they  consist  of  stars, 
and  which  any  increase  of  the  optical  power  of  the  telescope  may  be  ex- 
pected to  resolve  into  distinct  stars  j  3d.  Nebulae,  properly  so  called,  in 
which  there  is  no  appearance  whatever  of  stars ;  which,  again,  have  been 
subdivided  into  subordinate  uses,  according  to  their  brightness  and  size ; 

r.  r     '  See  also  Quarterly  iJe»ie»,  No.  94,  p.  540.  !'  1    - 


'  ill 

1 

1 

1  ■   1 

':'  t 

<'J 


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502 


OUTLINES  OF  ASTRONOMY. 


4th.  Planetary  nebulae ;  6tb.  Stellar  ncbuln;  and,  6th.  Nobuloas  Btan. 
The  great  power  of  his  telescope  disclosed  the  existence  of  an  immense 
number  of  these  objects  before  unknown,  and  showed  them  to  be  distri- 
buted  over  the  heavens,  not  by  any  means  uniformly,  but  with  a  marked 
preference  to  a  certain  district,  extending  over  the  northern  pole  of  the 
galactic  circle,  and  occupying  the  constellations  Leo,  Leo  Minor,  the  body, 
tail,  and  hind  legs  of  Ursa  Major,  Canes  Venatici,  Coma  Berenices,  the 
preceding  leg  of  Bootes,  and  the  head,  wing,  and  shoulders  of  Virgo.  In 
this  region,  occupying  about  one-eighth  of  the  whole  surface  of  the  sphere, 
one-third  of  the  entire  nebuU  us  contents  of  the  heavens  are  congregated. 
On  the  other  hand,  they  are  very  sparingly  scattered  over  the  constella- 
tions Aries,  Taurus,  the  head  and  shoulders  of  Orion,  Auriga,  Perseus, 
Camelopardalus,  Draco,  Hercules,  the  northern  part  of  Serpentarius,  the 
tail  of  Serpens,  that  of  Aquila,  and  the  whole  of  Lyra.  The  hours  3,  4, 
6,  and  16,  17,  18,  of  right  ascension  in  the  norther'i  hemisphere  are  sin- 
gularly poor,  and,  on  the  other  hand,  the  hours  10^  \^,  and  12  (but  espe- 
cially 12),  extraordinarily  rich  in  these  objects.  In  the  southern  hemi- 
sphere a  much  greater  uniformity  of  distribution  prevails,  and  with 
exception  of  two  very  remarkable  centres  of  accumulation,  called  the 
Magellanic  clouds  (of  which  more  presently),  there  is  no  very  decided 
tendency  to  their  assemblage  in  any  particular  region. 

(869.)  Clusters  of  stars  are  either  globular,  such  as  we  have  already 
described,  or  of  irregular  figure.  Tb  '■-.:  ?.  latter  are,  generally  speaking, 
less  rich  in  stars,  and  especially  less  condensed  towards  the  centre.  They 
are  also  less  definite  in  outline ;  so  that  it  is  often  not  easy  to  say  where 
they  terminate,  or  whether  they  are  to  be  regarded  otherwise  than  as 
merely  richer  parts  of  the  heavens  than  those  around  them.  Many, 
indeed,  the  greater  proportion  of  them,  are  situated  in  or  close  on  the 
borders  of  the  Milky  Way.  In  some  of  them  the  stars  are  nearly  all  of 
a  size,  in  others  extremely  different ;  and  it  is  no  uncommon  thing  to  find 
a  very  red  star  much  brighter  than  the  rest,  occupying  a  conspicuous 
situation  in  them.  Sir  William  Herschel  regards  these  as  globular  clus- 
ters in  a  less  advanced  state  of  condensation,  conceiving  all  such  groups 
as  approaching,  by  their  mutual  attrpcuon,  to  the  globular  figure,  and 
assembling  themselves  together  from  all  the  surrounding  region,  under 
laws  of  which  we  have,  it  is  true,  n<)  other  proof  than  the  observance  of 
a  gradation  by  which  their  characters  shade  into  one  another,  so  that  it  is 
impossible  to  say  where  one  species  ends  and  the  other  begins.  Among 
the  most  beautiful  objects  of  this  class  is  that  which  surrounds  the  star 
K  Crucis,  set  down  as  a  nebula  by  Lacaille.  It  occupies  an  area  of  about 
3ne  48th  part  of  a  square  degree,  and  consists  of  about  110  stars  from  the 


7th  mad 

coloured 

whole  til 

(870. 


RESOLVABLE   NEBULA. 


7th  magnitude  downwards,  eight  of  the  more  oonspicuous  of  which  are 
coloured  with  various  shades  of  red,  green,  and  blue,  so  m  to  give  to  thu 
whole  the  appearance  of  a  rich  piece  of  jewellery. 

(870.)  Resolvable  nebulce  can,  of  course,  only  be  considered  as  clusters 
either  too  remote,  or  consisting  of  stars  intrinsically  too  faint  to  a£fcot  us 
by  their  individual  light,  unless  where  two  or  three  happen  to  be  close 
enough  to  make  a  joint  impression,  and  give  the  idea  of  a  point  brighter 
than  the  rest.  They  are  almost  universally  round  or  oval  —  their  loose 
appendages,  and  irregularities  of  form,  being  as  it  were  extinguished  by 
the  distance,  and  the  only  general  figure  of  the  more  condensed  parts  being 
discernible.  It  is  under  the  appearance  of  objects  of  this  character  that 
all  the  greater  globular  clusters  exhibit  themselves  in  telescopes  of  insuffi- 
cent  optical  power  to  show  them  well ;  and  the  conclusion  is  obvious,  that 
those  which  the  most  powerful  can  barely  render  resolvable,  and  even 
those  which,  with  such  powers  as  are  usually  applied,  show  no  sign  of 
being  composed  of  stars,  would  be  completely  resolved  by  a  further  in* 
crease  of  optical  power.  In  fact,  this  probability  has  almost  been  con- 
verted into  a  certainty  by  the  magnificent  reflecting  telescope  constructed 
by  Lord  Kosse,  of  six  feet  in  aperture,  which  has  resolved  or  rendered 
resolvable  multitudes  of  nebulee  which  had  resisted  all  inferior  powers. 
The  sublimity  of  the  spectacle  afforded  by  that  instrument  of  some  of  the 
larger  globular  and  other  clusters  enumerated  in  the  list  given  in  Art.  867, 
is  declared  by  all  who  have  witnessed  it  to  be  such  as  no  words  can  express. 

(871.)  Although,  therefore,  nebulso  do  exist,  which  even  in  this  power- 
ful telescope  appear  as  nebulae,  without  any  sign  of  resolution,  it  may 
very  reasonably  be  doubted  whether  there  be  really  any  essential  physical 
distinction  between  nebula;  and  clusters  of  stars,  at  least  in  the  nature  of 
the  matter  of  which  they  consist,  and  whether  the  distinction  between 
such  nebulae  as  are  easily  resolved,  barely  resolvable  with  excellent  tele- 
scopes, and  altogether  irresolvable  with  the  best,  be  any  thing  else  than 
one  of  degree,  arising  merely  from  the  excessive  minuteness  and  multitude 
of  the  stars,  of  which  the  latter,  as  compared  with  the  former,  consist. 
The  first  impression  which  Halley,  and  other  early  discoverers  of  nebulous 
objects  received  from  their  peculiar  aspect,  so  different  from  the  keen, 
concentrated  light  of  mere  stars,  was  that  of  a  phosphorescent  vapour  (like 
the  matter  of  a  comet's  tail)  or  a  gaseous  and  (so  to  speak)  elementary  form 
of  luminous  sidereal  matter.'  Admitting  the  existence  of  such  a  medium, 
dispersed  in  some  cases  irregularly  through  vast  regions  in  space,  in  others 
confined  to  narrower  and  more  definite  limits,  Sir  W.  Herschel  was  led  tc 
speculate  on  its  gradual  subsidence  and  condensation  by  the  effect  of  its 

'  '•         '  •  Jialley,  Phil.  Trans-,  xxix.  p.  390. 


n.   I 

M  '•     8  ■ 


<  I 


■'i' 


Hi 


■  IM 


;*l 


604 


OUTLINES  OP  ASTRONOMY. 


own  gravity,  into  more  or  less  regular  spherical  or  flpbcroidul  forms, 
denser  (as  tboy  must  in  that  case  be)  towards  the  centre.  Assuming  that 
in  the  progress  of  this  subsidence,  local  centres  of  condensation,  subordi- 
nate to  the  general  tendency,  would  not  bo  wanting,  ho  cone*  ived  thnt  in 
this  way  solid  nuclei  might  arise,  whose  local  gravitation  still  further 
condensing,  and  so  absorbing  the  nebulous  matter,  each  in  its  immediate 
neighbourhood,  might  ultimately  become  stars,  and  tho  whole  ncbulm 
finally  take  on  the  state  of  a  cluster  of  stars.  Among  the  multitude  of 
nebulae  revealed  by  his  telescopes,  every  stage  of  this  process  might  bo 
considered  as  displayed  to  our  eyes,  and  in  every  modification  of  form  to 
which  the  general  principle  might  bo  conceived  to  apply.  The  more  or 
less  advanced  state  of  a  nebula  towards  its  segregation  into  discrete  stars 
and  of  these  stars  themselves  towards  a  denser  state  of  aggregation  round 
a  central  nucleus,  would  thus  be  in  some  sort  an  indication  of  age. 
Neither  is  there  any  variety  of  aspect  which  nebulae  offer,  which  stands  at 
all  in  contradiction  to  this  view.  Even  though  wo  should  feel  ourselves 
compelled  to  reject  the  idea  of  a  gaseous  or  vaporous  "  nebulous  matter," 
it  loses  little  or  none  of  its  force.  Subsidence,  and  the  central  aggrega- 
tion consequent  on  subsidence,  may  go  on  quite  as  well  among  a  multi- 
tude of  discrete  bodies  under  the  influence  of  mutual  attraction,  and 
feeble  or  partially  opposing  projectile  motions,  as  among  the  particles  of  a 
gaseous  fluid. 

(872.)  The  '^ nebular  hypothesis^*'  as  it  has  been  termed,  and  the 
heoi-y  of  sidereal  aggregation  stand,  in  fact,  quite  independent  of  each 
other,  the  one  as  a  physical  conception  of  processes  which  may  yet,  for 
aught  we  know,  have  formed  part  of  that  mysterious  chain  of  causes  and 
effects  antecedent  to  the  existence  of  separate  self-luminous  solid  bodies; 
the  other,  as  an  application  of  dynamical  principles  to  cases  of  a  very 
complicated  nature  no  doubt,  but  in  which  the  possibility  or  impossibility, 
at  least,  of  certain  general  results  may  be  determined  on  perfectly  legiti- 
mate principles.  Among  a  crowd  of  solid  bodies  of  whatever  size,  ani- 
mated by  independent  and  partially  opposing  impulses,  motions  opposite 
to  each  other  must  produce  collision,  destruction  of  velocity,  and  subsi- 
dence or  near  approach  towards  the  centre  of  preponderant  attraction; 
while  those  which  conspire,  or  which  remain  outstanding  after  such  con- 
flicts, must  ultimately  give  rise  to  circulation  of  a  permanent  character. 
Whatever  we  may  think  of  such  collisions  as  events,  there  is  nothing  in 
thi»  conception  contrary  to  sound  mechanical  principles.  It  will  be  recol- 
lected that  the  appearance  of  central  condensation  among  a  multitude  of 
separate  bodies  in  motion,  by  no  means  implies  permanent  proximity  to 
the  centre  in  each ;  an^  more  than  the  habitually  crowded  state  of  a 


markct- 
froqueut 
individii 
do  exist 
cally  ] 


THEORY   OF  TUB   FORMAtlON   OF   CH'STER8. 


605 


inarkct-p!,ic*,  ii  wbinb  a  hirgo  proportion  of  tho  iriLuhitants  of  a  tmn 
freijueully  or  occasionally  resort,  implies  the  pormonent  rt'sidcnf'e  of  each 
individual  within  its  area.  It  is  a  fact  that  clusters  thus  eontrally  crowded 
do  exist,  and  therefore  the  conditions  of  their  existence  must  be  dynanii- 
cally  I  )s,siblc,  and  in  what  has  been  said  we  may  at  least  perceive  some 
g1ini])scs  of  tho  manner  in  which  they  are  so.  The  actual  intervals  be- 
tween the  stars,  even  in  the  most  crowded  parts  of  a  resolved  uobula,  to 
be  seen  at  all  by  us,  must  bo  enormous.  Ages,  which  to  us  may  well 
appear  indefinite,  may  easily  be  conceived  to  pass  without  a  single  instance 
of  collision,  in  the  nature  of  a  catastrophe.  Such  may  have  gradually 
become  rarer  as  the  system  has  emerged  from  what  must  be  considered  its 
chaotic  state,  till  at  length,  in  the  fulness  of  time,  and  under  tho  pre- 
arranging guidance  of  that  Design  which  pervades  universal  nature,  each 
individual  may  have  taken  up  such  a  course  as  to  annul  the  possibility  of 
further  destruotiye  interference. 

(873.)  But  to  return  from  the  regions  of  speculation  to  the  description 
of  facts.  Next  in  regularity  of  form  to  the  globular  clusters,  whoso  con- 
sideration has  led  us  into  this  digression,  are  elliptic  nebulae,  more  or  less 
elongated.  And  of  these  it  may  be  generally  remarked,  as  a  foot  un- 
doubtedly connected  in  some  very  intimate  manner  with  tho  dynamical 
conditions  of  their  subsistence,  that  such  nebula)  are,  for  the  most  part, 
beyond  comparison  more  difficult  of  resolution  than  those  of  globular  form. 
They  ore  of  all  degrees  of  excentricity,  from  moderately  oval  forms  to 
ellipses  so  elongated  as  to  be  almost  linear,  which  are,  no  doubt,  edge- 
views  of  very  flat  ellipsoids.  In  all  of  them  the  density  increases  towards 
the  centre,  and  as  a  general  law  it  may  be  remarked  that,  so  far  as  we 
can  judge  from  their  telescopic  appearance,  their  internal  strata  approach 
more  nearly  to  the  spherical  form  than  their  external.  Their  rcsolva- 
bility,  too,  is  greater  in  the  central  parts,  whether  owing  to  a  real  ST\pe- 
riority  of  size  in  the  central  stars  or  to  the  greater  frequency  of  cases  of 
close  juxta-position  of  individuals,  so  that  two  or  three  united  appear  as 
one.  In  some  the  condensation  is  slight  and  gradual,  in  others  great  and 
sudden:  so  sudden,  indeed,  as  to  offer  the  appearance  of  a  dull  and 
blotted  star,  standing  in  the  midst  of  a  faint,  nearly  equable  elliptic  nebu- 
losity, of  which  two  remarkable  specimens  occur  in  R.  A.  12"  10"  33», 
N.  P.  D.  41°  46',  and  in  13"  27"  28',  119°  0'  (1830). 

(874.)  The  largest  and  finest  specimens  of  elliptic  nebulae  which  the 
heavens  afford  are  that  in  the  girdle  of  Andromeda  (near  the  star  v  of 
that  constellation)  and  that  discovered  in  1783,  by  Miss  Carolina  Herschel, 
in  R.  A.  0"  SO™  12»,  N.  P.  D.  116°  13'.  The  nebula  in  Andromeda 
(Plate  II.  fig.  3.)  is  visible  to  the  naked  eye,  and  is  continually  mistaken 


'   t 


v}M 


■i,  ( 


606 


OUTLINES  OF  ASTRONOMY. 


/ 


for  a  comet  by  those  unacquainted  with  the  heavens.  Simon  Marias, 
who  noticed  it  in  1612  (though  it  appears  also  to  have  been  seen  and 
described  as  oval,  in  995),  describes  its  appearance  as  that  of  a  candle 
shining  through  horn,  and  the  resemblance  is  not  inapt.  Its  form,  as  seen 
through  ordinary  telescopes,  is  a  pretty  long  oval,  increasing  by  insensible 
gradations  of  brightness,  at  first  very  gradually,  but  at  last  more  rapidly, 
up  to  a  central  point,  which,  though  very  much  brighter  than  the  rest,  is 
decidedly  not  a  star,  but  nebula  of  the  same  general  character  with  the 
rest  in  a  state  of  extreme  condensation.  Casual  stars  are  scattered  over 
it,  but  with  a  reflector  of  18  inches  in  diameter,  there  is  nothing  to  excite 
any  suspicion  of  its  consisting  of  stars.  Examined  with  instruments  of 
superior  defining  power,  however,  the  evidence  of  its  resolvability  into 
stars,  may  be  regarded  as  decisive.  Mr.  Gt.  P.  Bond,  assistant  at  the 
observatory  of  Cambridge,  U.  S.,  describes  and  figures  it  as  extending 
nearly  2i°  in  length,  and  upwards  of  a  degree  in  breadth  (so  as  to  include 
two  other  smaller  adjacent  nebulas),  of  a  form,  generally  speaking,  oval, 
but  with  a  considerably  protuberant  irregularity  at  its  north  following  ex- 
tremity, very  suddenly  condensed  at  the  nucleus  almost  to  the  semblance 
of  a  star,  and  though  not  itself  clearly  resolved,  yet  thickly  sown  over 
with  visible  minute  stars,  so  numerous  as  to  allow  of  200  being  counted 
within  a  field  of  20'  diameter  in  the  richest  parts.  But  the  most  remark- 
able teature  in  his  description  is  that  of  two  perfectly  straight,  narrow, 
and  comparatively  or  totally  obscure  streaks  which  run  nearly  the  whole 
length  of  one  side  of  the  nebula,  and  (though  slightly  divergent  from 
each  other)  nearly  parallel  to  its  longer  axis.  These  streaks  (which 
obviously  indicate  a  stratified  structure  in  the  nebula,  if,  indeed,  they  do 
not  originate  in  the  interposition  of  imperfectly  transparent  matter  between 
us  and  it)  are  not  seen  on  a  general  and  cursory  view  of  the  nebula ;  they 
require  attention  to  distinguish  them,*  and  this  circumstance  must  be  borne 
in  mind  when  inspecting  the  very  extraordinary  engraving  which  illustrates 
Mr.  Bond's  account.  The  figure  given  in  our  Plate  II.  fig.  3,  is  from  a 
rather  hasty  sketch,  and  makes  no  pretensions  to  exactness.  A  similar, 
but  much  more  strongly  marked  case  of  parallel  arrangement  than  that 
noticed  by  Mr.  Bond  in  this,  is  one  in  which  the  two  semi-ovals  of  an 
clliptically  formed  nebula  appear  cut  asunder  and  separated  by  a  broad 
obscure  band  parallel  to  the  larger  axis  of  the  nebula,  in  the  midst  of 
which  a  faint  streak  of  light  parallel  to  the  sides  of  the  cut  appears,  is 
seen  in  the  southern  hemisphere  in  R.  A.  13"  15"  31',  N.  P.  D.  132°  8' 


'  Account  of  the  nebula  in  Andromeda,  by  G.  P.  Bond,  Assistant  at  the  Cambridge 
Observatory,  U.  8.    Trans.  American  Acad.,  vol.  iii.  p.  80. 


)pears,  is 
132°  8' 


PLANBTART  NEBULiB. 


607 


(1830).    The  nebulae  in  12"  27-  3',  63"  5',  and  12«'  31-  11*,  100°  40' 
present  analogous  features. 

(875.)  Annular  nebulae  also  exist,  but  are  among  the  rarest  objects  in 
the  heavens.  The  most  conspicuous  of  this  class  is  to  be  found  almost 
exactly  half  way  between  0  and  y  Lyrae,  and  may  be  seen  with  a  telescope 
of  moderate  power.  It  is  small  and  particularly  well  defined,  so  as  to 
have  more  the  appearance  of  a  flat  oval  solid  ring  than  of  a  nebula.  The 
axes  of  the  ellipse  are  to  each  other  in  the  proportion  of  about  4  to  5,  and 
the  opening  occupies  about  half  or  rather  more  than  half  the  diameter. 
The  central  vacuity  is  not  quite  dark,  but  is  filled  in  with  faint  nebula, 
like  a  gauze  stretched  over  a  hoop.  The  powerful  telescopes  of  Lord 
Kosse  resolve  this  object  into  excessively  minute  stars,  and  show  filaments 
of  stars  adheripg  to  its  edges.* 

(876.)  Planetary  nebul.^  are  very  extraordinary  objects.  They 
have,  as  their  name  imports,  a  near,  in  some  instances,  a  perfect  resem- 
blance to  planets,  presenting  discs  round,  or  slightly  oval,  in  some  quite 
sharply  terminated,  in  others  a  little  hazy  or  softened  at  the  borders. 
Their  light  is  in  some  perfectly  equable,  in  others  mottled  and  of  a  very 
peculiar  texture,  as  if  curdled.  They  are  comparatively  rare  objects,  not 
above  four  or  five  and  twenty  having  been  hitherto  observed,  and  of  these 
nearly  three-fourths  are  situated  in  the  southern  hemisphere.  Being  very 
interesting  objects,  we  subjoin  a  list  of  the  most  remarkable.'  Among 
these  may  be  more  particularly  specified  the  sixth  in  order,  situated  in  the 
Cross.  Its  light  is  about  equal  to  that  of  a  star  of  the  6*7  magnitude, 
its  diameter  about  12",  its  disc  circular  or  very  slightly  elliptic,  and  with 
a  clear,  sharp,  well-defined  outline,  having  exactly  the  appearance  of  a 
planet  with  the  exception  only  of  its  colour,  which  is  a  fine  and  full  blue 
verging  somewhat  upon  green.  And  it  is  not  a  little  remarkable  that  this 
phsenomenon  of  a  blue  colour,  which  is  so  rare  among  stars  (except  when 
in  the  immediate  proximity  of  yellow  stars)  occurs,  though  less  strikingly, 
in  three  other  objects  of  this  class,  viz.  in  No.  4,  whose  colour  is  sky-blue. 


'  The  places  of  the  annular  nebulae,  at  present  known  (for  1830)  are, 


1. 
2. 


17h 
17 


R.  A. 

10"     39' 
19         2 


N.  P.  D. 
128<»  18' 
113 


3. 
4. 


20 


R.A. 

47» 
9 


13« 
33 


N.  P.D. 

57°      11' 
59       57 


*  Places  for  1830  of  twelve  of  the  most  remarkable  planetary  nebulae. 


R.A. 

N.  P.  D. 

R.A. 

N.  P.  D. 

R.A. 

N.  P.  D. 

h.  m.  8. 
1.  7  34  2 
1  9  16  .39 

3.  9  59  52 

4.  10  16  36 

o   / 

104  29 
147  35 
129  36 

107  47 

h.  m.  8. 

5.  11  4  49 

6.  11  41  56 

7.  15  5  18 

8.  19  10  9 

o   / 

34  4 
146  14 
135  1 

83  46 

h.  m.  8. 
9.  19  34  21 

10.  19  40  19 

11.  20  54  53 

12.  23  17  44 

o   / 

104  33 
39  54 

102  2 
48  24 

i^iiM 


m  ■ 


IP' 


,1 "  '■ 


-i! 


u 

i 

"'! 

'i 

1 

i 

ft^ 

if 

I  mi 


ii 

! 

>Uii. 

% 

m. 

\mk 

608 


OUTLINES   OP  ASTRONOMY. 


and  in  Nos.  11  and  12,  where  the  tint,  though  paler,  is  still  evident. 
Nos.  2,  7,  9,  and  12,  are  also  exceedingly  characteristic  objects  of  this 
class.  Nos.  3,  5,  and  11  (the  latter  in  tae  parallel  of  »  Aquarii,  and 
about  5"  preceding  that  star),  are  considerably  elliptic,  and  (respectively) 
about  38",  30"  and  15"  in  diameter.  On  the  disc  of  No  3,  and  very 
nearly  in  the  centre  of  the  ellipse,  is  a  star  9",  and  the  texture  of  its  light, 
being  velvety,  or  as  if  formed  of  fine  dust,  clearly  indicates  its  resolvability 
into  stars.  The  largest  of  these  objects  is  No.  5,  situated  somewhat  south 
of  the  parallel  of  j3  Ursse  Majoris  and  about  12"  following  that  star.  Its 
apparent  diameter  is  2'  40",  which,  supposing  it  placed  at  a  distance  from 
us  not  more  than  that  of  61  Cygni,  would  imply  a  linear  one  seven  times 
greater  than  that  of  the  orbit  of  Neptune.  The  light  of  this  stupendous 
globe  is  perfectly  equable  (except  just  at  the  edge  whete  it  is  slightly 
softened),  and  of  considerable  brightness.  Such  an  appearance  would  not 
be  presented  by  a  globular  space  uniformly  filled  with  stars  or  luminous 
matter,  which  structure  would  necessarily  give  rise  to  an  apparent  increase 
of  brightness  towards  the  centre  in  proportion  to  the  thickness  traversed 
by  the  visual  ray.  We  might,  therefore,  be  induced  to  conclude  its  real 
constitution  to  be  either  that  of  a  hollow  spherical  shell  or  of  a  flat  disc, 
presented  to  us  (by  a  highly  improbable  coincidence)  in  a  plane  precisely 
perpendicular  to  the  visual  ray. 

(877.)  Whatever  idea  we  may  form  of  the  real  nature  of  such  a  body, 
or  of  the  planetary  nebulae  in  general,  which  all  agree  in  the  absence  of 
central  condensation,  it  is  evident  that  the  intrinsic  splendour  of  their 
surfaces,  if  continuous,  must  be  almost  infinitely  less  than  that  of  the 
sun.  A  circular  portion  of  the  sun's  disc,  subtending  an  angle  of  1', 
would  give  a  light  equal  to  that  of  780  full  moons ;  while  among  all  the 
objects  in  question  there  is  not  one  which  can  be  seen  with  the  naked  eye. 
M.  Arago  has  surmised  that  they  may  possibly  be  envelopes  shining  by 
reflected  light,  from  a  solar  body  placed  in  their  centre,  invisible  to  us  by 
the  effect  of  its  excessive  distance  j  removing,  or  attempting  to  remove 
the  apparent  paradox  of  such  an  explanation,  by  the  optical  principle  that 
an  illuminated  surface  is  equally  hrighi  at  all  distances,  and,  therefore,  if 
large  enough  to  subtend  a  measurable  angle,  can  be  equally  well  seen, 
whereas  the  central  body,  subtending  no  such  angle,  has  its  effect  on  our 
sight  diminished  in  the  inverse  ratio  of  the  square  of  its  distance.'    The 


'  With  duo  deference  to  so  high  an  authority  we  must  demur  to  the  conclusion. 
Even  supposing  the  envelope  to  reflect  and  scatter  (equnlly  in  all  directions)  all  the 
light  of  the  central  sun,  the  portion  of  the  light  so  scattered  which  would  fall  to  our 
share,  could  not  exceed  that  which  that  sup  itself  would  send  to  us  by  direct  radiation. 
But  this,  ex  hypothesi,  is  too  small  to  affect  the  eye  with  any  luminous  perception,  much 


I  I'-^^H 


p  1 


DOUBLE   NEBULA. 


609 


assiduous  application  of  the  immense  optical  powers  recently  brought  to 
bear  on  the  heavens,  will  probably  remove  some  portion  of  the  mystery 
which  at  present  hangs  about  these  enigmatical  objects. 

(878.)  Double  nebulae  occasionally  occur — and  when  such  is  the  case, 
the  constituents  most  commonly  belong  to  the  class  of  spherical  nebulae, 
and  are  in  some  instances  undoubtedly  globular  clusters.    All  the  varieties 
of  double  stars,  in  fact,  as  to  distance,  position,  and  relative  brightness, 
have  their  counterparts  in  double  nebulae ;  besides  which  the  varieties  of 
form  and  gradation  of  light  in  the  latter  aflFord  room  for  combinations 
peculiar  to  this  class  of  objects.     Though  the  conclusive  evidence  of  ob- 
served relative  motion  be  yet  wanting,  and  though  from  the  vast  scale  on 
which  such  systems  are  constructed,  and  the  probable  extreme  slowness 
of  the  angular  motion,  it  may  continue  foi  ages  to  be  so,  yet  it  is  impos- 
sible, when  we  cast  our  eyes  upon  such  objects,  or  on  the  figures  which 
have  been  given  of  them,'  to  doubt  their  physical  connexion.     The  argu- 
ment drawn  from  the  comparative  rarity  of  the  objects  in  proportion  to 
the  whole  extent  of  the  heavens,  so  cogent  in  the  case  of  the  double  stars, 
is  infinitely  more  so  in  that  of  the  double  nebulae.     Nothing  more  magni- 
ficent can  be  presented  to  our  consideration,  than  such  combinations. 
Their  stupendous  scale,  the  multitude  of  individuals  they  involve,  the 
perfect  symmetry  and  regularity  which  many  of  them  present,  the  utter 
disregard  of  complication  in  thus  heaping  together  system  upon  system, 
and  construction  upon  construction,  leave  us  lost  in  wonder  and  admira- 
tion at  the  evidence  they  afford  of  infinite   power  and  unfathomable 
design. 

(879.)  Nebulae  of  regular  forms  often  stand  in  marked  and  symmetrical 
relation  to  stars,  both  single  and  double.  Thus  we  are  occasionally  pre- 
sented with  the  beautiful  and  striking  phaenomenon  of  a  sharp  and  bril- 
liant star  concentrically  surrounded  by  a  perfectly  circular  disc  or  atmo- 
sphere of  faint  light,  in  some  cases  dying  away  insensibly  on  all  sides,  in 
others  almost  suddenly  terminated.  These  are  Nebulous  Stars.  Fine 
examples  of  this  kind  are  the  45th  and  69th  nebulae  of  Sir  Wm.  Her- 
schel's  fourth  class«  (R.  A.  7"  19-'  8%  N.  P.  D.  68''  45',  and  3"  58-  36% 

less  then  could  it  do  so  if  spread  over  a  surface  many  million  times  exceeding  in  angular 
area  the  apparent  disc  of  the  central  sun  itself.  (See  Annuaire  du  Bureau  des  Longi- 
tudes, 1842,  p.  409,  410,  411.)  M.  Arago  is  expressly  contending  for  reflected  light. 
If  the  envelope  be  self-luminous,  his  reasoning  is  perfectly  sound. 

•  Phil.  Trans.,  1833.    Plate  vii. 

'The  classes  here  referred  to  are  not  the  species  described  in  Art.  868,  but  lists  of 
nebulae,  eight  in  number,  arranged  according  to  brightness,  size,  dentity  of  clustering, 
&c.,  in  one  or  other  of  which  all  nebulae  were  originally  classed  by  him.  Class  I. 
contains  "  Bright  nebulffi ;"  !I.  "  Faint  do. ;"  III.  "  Very  faint  do. ;"  IV.  "  Planetary 


'i 

H   I 


M 


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i 
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t 
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di 

510 


OUTLINES   OP  ASTRONOMY. 


59°  40'),  in  which  stars  of  the  8th  magnitude  are  surrounded  by  photo- 
spheres of  the  kind  above  described  respectively  of  12"  and  25"  in  dia- 
meter. Among  stars  of  larger  magnitudes,  55  Andromedse  and  8  Canum 
Yenatioorum  niay  be  named  as  exhibiting  the  same  phaenomenon  with 
more  brilliancy,  but  perhaps  with  less  perfect  regularity. 

(880.)  The  connexion  of  nebulso  with  double  stars  is  in  many  instances 
extremely  remarkable.  Thus  in  R.  A.  IS"*  7-  1',  N.  P.  D.  109°  56', 
occurs  an  elliptic  nebula  having  its  longer  axis  about  50",  in  which,  sym- 
metrically placed,  and  rather  nearer  the  vertices  than  the  foci  of  the  ellipse, 
are  the  equal  individuals  of  a  double  star,  each  of  the  10th  magnitude. 
In  a  similar  combination  noticed  by  M.  Strove  (in  R.  A.  18"  25",  N.  P.  D. 
25°  7'),  the  stars  are  unequal  and  situated  precisely  at  the  two  extremities 
of  the  major  axis.  In  R.  A.  13"  47-  33»,  N.  P.  D.  129°  9',  an  oval 
nebula  of  2'  in  diameter  has  very  near  its  centre  a  close  double  star,  the 
individuals  of  which,  slightly  unequal,  and  aVyxxt  the  9-10  magnitude,  are 
not  more  than  2"  asunder.  The  nucleus  of  Messier's  64  th  nebula  is 
'•  strongly  suspected"  to  be  a  close  double  star — and  several  other  instances 
might  be  cited. 

(881.)  Among  the  nebulae  which,  though  deviating  more  from  sym- 
metry of  form,  are  yet  not  wanting  in  a  certain  regularity  of  figure,  and 
which  seem  clearly  entitled  to  be  regarded  as  systems  of  a  definite  nature, 
however  mysterious  their  structure  and  destination,  by  far  the  most  re- 
markable are  t^e  27th  and  51st  of  Messier's  Catalogue.'  This  consists 
of  two  round  or  somewhat  oval  nebulous  masses  united  by  a  short  neck 
of  nearly  the  same  density.  Both  this  and  the  masses  graduate  off  how- 
ever into  a  fainter  nebulous  envelope  which  completes  the  figure  into  an 
elliptic  form,  of  which  the  interior  masses  with  their  connexion  occupy  the 
lesser  axis.  Seen  in  a  reflector  of  18  inches  in  aperture,  the  form  has 
considerable  regularity ;  and  though  a  few  stars  are  here  and  there  scat- 
tered over  it,  it  is  unresolved.  Lord  Rosse,  viewing  it  with  a  reflector  of 
double  that  aperture,  describes  and  figures  it  as  resolved  into  numerous 
stars  with  much  intermixed  nebula ;  while  the  symmetry  of  form  by  ren- 
dering visible  features  too  faint  to  be  seen  with  inferior  power,  is  rendered 
considerably  less  striking,  though  by  no  means  obliterated. 

(882.)  The  51st  nebula  of  Messier,  viewed  through  an  18-inch  re- 
flector, presents  the  appearance  of  a  large  and  bright  globular  nebula, 


nebulHB,  stars  with  bars,  milky  chevelures,  short  rays,  remaritable  shapes,  &c. ;"  V. 
"  V-jry  large  nebulae;"  VI.  "  Very  comDressed  rich  clusters;"  VII.  "  Pretty  much 
rumpressed  do.;"  VIII.  "  Coarsely  scattered  clusters." 

'  Place  for  1830:  R.  A.  19"  52°'  12',  N.  P.  D.  67°  44',  and  R.  A.  13"  22™  39;  N.  P. 
D.  41°56'. 


NEBULJB   OF   PECULIAR   FORMS. 


611 


surrounded  by  a  ring  at  a  considerable  distance  from  the  globe,  very  une- 
qual in  brightness  in  its  different  parts,  and  subdivided  through  about  two- 
fifths  of  its  circumference  as  if  into  two  laminae,  one  of  which  appears 
as  if  turned  up  towards  the  eye  out  of  the  plane  of  the  rest.  Near  it 
(at  about  a  radius  of  the  ring  distant)  is  a  small  bright  round  nebula. 
Viewed  through  the  6-feet  reflector  of  Lord  Rosse  the  aspect  is  much 
altered.  The  interior,  or  what  appeared  the  upturned  portion  of  the  ring, 
assumes  the  aspect  of  a  nebulous  coil  or  convolution  tending  in  a  spiral 
form  towards  the  centre,  and  a  general  tendency  to  a  spiroid  arrangement 
of  the  streaks  of  nebula  connecting  the  ring  and  central  mass  which  this 
power  brings  into  view,  becomes  apparent,  and  forms  a  very  striking 
feature.  The  outlying  nebula  is  also  perceived  to  b'j  connected  by  a 
narrow,  curved  band  of  nebulous  light  with  the  ring,  and  the  whole,  if 
not  clearly  resolved  into  stars,  has  a  "  resolvable"  character  which  evi- 
dently indicates  its  composition.* 

(883.)  We  come  now  to  a  class  of  nebulae  of  totally  different  character. 
They  are  of  a  very  ^eat  extent,  utterly  devoid  of  all  symmetry  of  form, 
—  on  the  contrary,  irregular  and  capricious  in  their  shapes  and  convolu- 
tions to  a  most  extraordinary  degree,  and  no  less  so  in  the  distribution  of 
their  light.  No  two  of  them  can  be  said  to  present  any  similarity  of 
figure  or  aspect,  but  they  have  one  important  character  in  common. 
They  are  all  situated  in,  or  very  near,  the  borders  of  the  Milky  Way. 
The  most  remote  from  it  is  that  in  the  sword  handle  of  Orion,  which 
being  20"  from  the  galactic  circle,  and  15°  from  the  visible  border  of  the 
Via  Lactea,  might  seem  to  form  an  exception,  though  not  a  striking  one. 
But  this  very  situation  may  be  adduced  as  a  corroboration  of  the  general 
view  which  this  principle  of  localization  suggests.  F( .  the  place  in  ques- 
tion is  situated  in  the  prolongation  of  that  faint  offset  of  the  Milky  Way 
which  we  traced  (Art.  787.)  from  o  and  c  Persei  towards  Aldebaran  and 
the  Hyades,  and  in  the  zone  of  Great  Stars  noticed  in  Art.  785.  as  an 
appendage  of,  and  probably  bearing  relation  to  that  stratum. 

(884.)  From  this  it  would  appear  to  follow,  almost  as  a  matter  of 
course,  that  they  must  be  regarded  as  outlying,  very  distant,  and  as  it 
were  detached  fragments  of  the  great  stratum  of  the  Galaxy,  and  this 
view  01  tli8  subject  is  strengthened  when  we  find  on  mapping  down  their 
places  that  they  may  all  be  grouped  in  four  great  masses  or  nebulous 
regions,  —  that  of  Orion,  of  Argo,  of  Sagittarius,  and  of  Cygnus.  And 
thus,  inductively,  we  may  gather  some  information  respecting  the  struc- 

'  This  description  is  from  the  recoUectiMn  of  a  sketch  exhibited  by  his  I<ordship  at 
the  British  Association.  Every  astronomer  must  long  for  the  publication  of  his  own 
account  of  the  wonders  disclosed  by  this  magnificent  instrument. 


'  I 


Ih 


f  1 
4 


,1  ><■ 


''fe    ^ 


ti: 


liii! 


612 


OUTLINES   OP   ASTRONOMY. 


ture  and  form  of  the  Galaxy  itself,  which,  could  we  view  it  as  a  whole, 
from  a  distance  such  as  that  which  separates  us  from  these  objects,  would 
very  probably  present  itself  under  an  aspect  quite  as  complicated  and 
irregular. 

(885.)  The  great  nebula  surrounding  the  stars  marked  0  1  in  the  sword 
handle  of  Orion  was  discovered  by  Huyghens  in  1656,  and  has  been  re- 
peatedly figured  and  descrihed  by  astronomers  since  that  time.  Its 
appearance  varies  greatly  (as  that  of  all  nebulous  objects  does)  with  the 
instrumental  power  applied,  so  that  it  is  difficult  to  recognize  in  represent- 
ations made  with  inferior  telescopes,  even  principal  features,  to  say 
nothing  of  subordinate  details.  Until  this  became  well  understood,  it 
was  supposed  to  have  changed  very  materially,  bo-h  in  form  and  extent, 
during  the  interval  elapsed  since  l.s  first  discovery.  No  doubt,  however, 
now  remains  that  these  supposed  changes  have  originated  partly  from  the 
cause  above-mentioned,  partly  from  the  difficulty  of  correctly  drawing, 
and,  still  more,  engraving  such  objects,  and  partly  from  a  want  of  suffi- 
cient care  in  the  earlier  delineators  themselves  in  faithfully  copying  that 
which  they  really  did  see.  Our  figure  (Plate  IV.,  fig.  1,)  is  reduced 
from  a  larger  one  made  under  very  favourable  circumstances,  from  draw- 
ings taken  with  an  18-inch  reflector  at  the  Cape  of  Good  Hope,  where  its 
meridian  altitude  greatly  exceeds  what  it  has  at  Euro^)ean  stations.  The 
area  occupied  by  this  figure  is  about  one  25th  part  of  a  square  degree, 
extending  in  R.  A.  (or  horizontally)  2"  of  time,  equivalent  almost  ex- 
actly to  30'  in  arc,  the  object  being  very  near  the  equator,  and  24'  verti- 
cally, or  in  polar  distance.  The  figure  shows  it  reversed  in  both  direc- 
tions, the  northern  side  being  lowermost,  and  the  preceding  towards  the 
left  hand.  In  form,  the  brightest  portion  offers  a  resemblance  to  the  head 
and  yawning  jaws  of  some  monstrous  animal,  with  a  sort  of  proboscis  run* 
ning  out  from  the  snout.  Many  stars  are  scattered  over  it,  which  for  the 
most  part  appear  to  have  no  connexion  with  it,  and  the  remarkable  sex- 
tuple star  B  1  Orionis,  of  which  mention  has  already  been  made  (Art. 
837),  occupies  a  most  conspicuous  situation  close  to  the  brightest  portion, 
at  almost  the  edge  of  the  opening  of  the  jaws.  It  is  remarkable,  how- 
ever, that  within  the  area  of  the  trapezium  no  nebula  exists.  The  general 
aspect  of  the  less  luminous  and  cirrous  portion  is  simply  nebulous  and 
irresolvable,  but  the  brighter  portion  immediately  adjacent  to  the  trape- 
zmm,  forming  the  square  front  of  the  head,  is  shown  with  the  18-inch 
reflector  broken  up  into  masses  (very  imperfectly  represented  in  the  figure), 
whose  mottled  and  curdling  light  evidently  indicates  by  a  sort  of  granular 
texture  its  consisting  of  stars,  and  when  examined  under  the  great  light 
of  Lord  Rosse's  reflector,  or  the  exquisite  defining  power  of  the  great 


NEBULA   OF   ARGO. 


613 


Achromatic  at  Cambridge,  U.  S.,  is  evidently  perceived  to  consist  of  clus- 
tering stars.  There  can  thercibre  be  little  doubt  as  to  the  whole  consist- 
ing of  stars,  too  minute  to  be  discerned  individually  even  with  these 
powerful  aids,  but  which  become  visible  as  points  of  light  when  closely 
adjacent  in  the  more  crowded  parts  in  the  mode  already  more  than  once 
suggested. 

(886.)  The  nebula  is  not  confined  to  the  limits  of  our  figure.  North- 
ward of  9  about  38',  and  nearly  on  the  same  meridian  are  two  stars 
marked  C  1  and  C  2  Orionis,  involved  in  a  bright  and  branching  nebula 
of  very  singular  form,  and  south  of  it  is  the  star  (  Orionis,  which  is  also 
involved  in  strong  nebula.  Careful  examination  with  powerful  telescopes 
has  traced  out  a  continuity  of  nebulous  light  between  the  great  nebula 
and  both  these  objects,  and  there  can  be  little  doubt  that  the  nebulous 
region  extends  northwards,  as  far  as  e  in  the  belt  of  Orion,  which  is  in- 
volved in  strong  nebulosity,  as  well  as  several  smaller  stars  in  the  immedi- 
ate neighbourhood.  Professor  Bond  has  given  a  beautiful  figure  of  th** 
great  nebula  in  Trans.  American  Acad,  of  Arts  and  Sciences,  new  series, 
vol.  "i. 

(d87.)  The  remarkable  variation  in  lustre  of  the  bright  star  j^  in  Argo, 
has  been  already  mentioned.  This  star  is  situated  in  the  most  condensed 
region  of  a  very  extf  osive  nebula  or  congeries  of  nebular  masses,  streaks 
and  branches,  a  portion  of  which  is  represented  in  fig.  2,  Plate  IV.  The 
whole  nebula  is  spread  over  an  area  of  fully  a  square  degree  in  extent, 
of  which  that  included  in  the  figure  occupies  about  one-fourth,  that  is  to 
say,  28'  in  pc'iar  distance,  and  32'  of  arc  in  R.  A.,  the  portion  not  in- 
cluded being,  though  fainter,  even  more  capriciously  contorted  than  that 
here  depicted,  in  which  it  should  be  observed  that  the  preceding  side  is 
towards  the  right  band,  and  the  southern  uppermost.  Viewed  with  an 
i8-inch  reflector,  no  part  of  this  strange  object  shows  any  sign  of  resolu- 
tion into  stars,  nor  in  the  brightest  and  most  condensed  portion  adjacent 
to  the  singular  oval  vacancy  in  the  middle  of  the  figure  is  there  any  of 
that  curdled  appearance,  or  that  tendency  to  break  up  into  bright  knots 
with  intervening  darker  portions  which  characisrize  the  nebula  of  Orion, 
and  indicate  its  resolvability.  The  whole  is  situated  in  a  very  rich  and 
brilliant  part  of  the  Milky  Way,  so  thickly  strewed  with  stars  (omitted 
in  the  figure),  that  in  the  area  occupied  by  the  nebula,  not  less  than  1200 
have  been  actually  cout  'ed,  and  their  places  in  R.  A.  and  P.  D.  deter- 
mined. Yet  it  is  obvious  that  these  have  no  connexion  whatever  with 
the  nebula,  being,  in  fact,  only  a  simple  continuation  over  it  of  the  general 
ground  of  the  galaxy,  which  on  an  average  of  two  hours  in  Right  Ascen- 
sion in  this  period  of  its  course,  contains  ao  less  than  3188  stars  to  th« 
33 


If 


il 


!-H 


HH 


514 


OUTLINES   OP  ASTRONOMY. 


square  degree,  oil,  however,  distinct,  and  (except  where  the  object  in 
question  is  situated)  seen  projected  on  u  perfectly  dark  heaven,  without 
any  appearance  of  intern>ixod  nebulosity.  The  conclusion  can  hardly  be 
avoided,  that  in  looking  at  it  we  see  through,  and  beyond  the  Milky  Way, 
fur  out  into  space,  through  a  starless  region,  disconnecting  it  altogether 
from  our  system.  "  It  is  not  easy  for  language  to  convey  a  full  impres- 
sion of  the  beauty  and  sublimity  of  the  spectacle  which  this  nebula 
offers,  as  it  enters  the  field  of  view  of  a  telescope  fixed  in  Right  Ascen- 
sion, by  the  diurnal  motion,  ushered  in  as  it  is  by  so  glorious  and  innu- 
merable a  procession  of  stars,  to  which  it  forms  a  sort  of  climax,"  and  in 
a  part  of  the  hDavens  otherwise  full  of  interest.  One  other  bright  and 
very  remarkably  formed  nebula  of  considerable  magnitude  precedes  it 
nearly  on  the  same  parallel,  but  without  any  traceable  connexion  between 
them.  '■'    ■ 

(888.)  The  nebulous  gi  jup  of  Sagittarius  consists  tf  several  conspicuous 
nebulae'  of  very  extraordinary  forms,  by  no  means  easy  to  give  an  idea  of 
by  mere  description.  One  of  them  (h,  1991')  is  singularly  trifid,  con- 
sisting of  three  bright  and  irregularly  formed  nebulous  masses,  graduating 
away  insensibly  externally,  but  coming  up  to  a  great  intensity  of  light  at 
their  interior  edges,  where  they  enclose  and  surround  a  sort  of  three-forked 
rift,  or  vacant  area,  abruptly  and  uncouthly  crooked,  and  quite  void  of 
nebulous  light.  A  beautiful  triple  star  ia  situated  precisely  on  the  edge 
of  one  of  these  nebulous  masses  just  where  the  interior  vacancy  forks  out 
two  channels.  A  fourth  nebulous  mass  spreads  like  a  fan  or  downy  plume 
from  a  star  at  a  little  distance  from  the  triple  nebula. 

(889.)  Nearly  adjacent  to  the  last  described  nebub,  and  no  doubt  con- 
nected with  it,  though  the  connexion  has  not  yet  been  traced,  is  situated 
the  8th  nebula  of  M€S8ier*s  Catalogue.  It  is  a  collection  of  nebulous 
folds  and  masses,  surrounding  and  including  a  number  of  oval  dark  vacan- 
cies, and  in  one  place  coming  up  to  so  great  a  degree  of  brightness,  as  to 
offer  the  appearance  of  an  elongated  nucleus.  Superposed  upon  this 
nebula,  and  extending  in  one  direction  beyond  its  area,  is  a  fine  and  rich 
cluster  of  scattered  stars,  which  seem  to  have  no  connexion  with  it,  as  the 

'  About  R.A.  17*  S?.",  N.P.D.  113°  1',  four  nebulm.  No.  41  of  Sir  Wm.  Herschel's 
4th  class,  and  Nos.  1,  2,  3,  of  his  5th,  uU  connected  into  one  great  complex  nebula.— 
In  R.A.  17"  53»  27*,  N.P.L.  114°  21',  the  S'.h,  and  in  18"  11",  106°  15',  the  17th  of 
Messier's  Catalogue. 

•  This  number  refers  to  the  catalogue  c*"  nebulae  in  Phil.  Trans.,  1833.  The  reader 
will  find  figures  of  the  several  nebulae  of  this  group  in  that  volume,  pi.  iv.,  fig.  35,  in  the 
Author's  '•  Results  of  Observations,  &.c.,  at  the  Cape  of  Good  Hope,"  Plates  i.  fig.  1, 
and  ii.  figs.  1  and  2,  and  in  Mason's  Memoir  in  the  collection  of  the  American  Phil.  Soc. 
vol.  vii.  art.  xiii. 


THE   MAOBLLANIO   CLOUDS. 


515 


nebula  dots  not,  as  in  the  region  of  Orion,  show  any  tendency  to  congre- 
gate about  the  stars. 

(890.)  '.rhe  19th  ne>ula  of  Mossier's  Catalogue,  though  some  degrees 
remote  from  the  oth(  .d,  evidently  belongs  to  this  group.  Its  form  is  very 
remarkable,  consisting  of  two  loops  like  capital  Greek  Omegas,  the  one 
bright,  the  other  exceedingly  faint,  connected  at  their  bases  by  a  broad 
and  very  bright  band  of  nebula,  insulated  within  which  by  a  narrow 
comparatively  obscure  border,  stands  a  bright,  resolvable  knot,  or  what  is 
probably  a  cluster  of  exceedingly  minute  stars.  A  very  faint  round  nebula 
stands  in  connexion  with  the  upper  or  convex  portion  of  the  brighter  loop. 
(891.)  The  nebulous  group  of  Cygnus  consists  of  several  largo  and 
irregular  nebulse,  one  of  which  passes  through  the  double  star  k  Cygni, 
as  a  long,  crooked  narrow  streak,  forking  out  in  two  or  three  places.  The 
others,'  observed  in  the  first  instance  by  Sir  W.  Hersohel  and  by  the 
author  of  this  work  as  separate  nebulas,  have  been  traced  into  connexion 
by  Mr.  Mason,  and  shown  to  form  part  of  a  curious  and  intricate  nebulous 
B^stem,  consisting,  1st,  of  a  long,  narrow,  curved,  and  forked  streak,  and 
2dly,  of  a  cellular  effusion  of  great  extent,  in  which  the  nebula  occurs 
intermixed  with,  and  adhering  to  stars  around  the  borders  of  the  cells, 
while  their  interior  is  free  from  nebula,  and  almost  so  from  stars. 

(892.)  The  Magellanic  clouds,  or  the  nubecula)  (major  and  minor,)  as 
they  are  called  in  the  celestial  maps  and  charts,  are,  as  their  name  imports, 
two  nebulous  or  cloudy  masses  of  light,  conspicuously  visible  to  the  naked 
eye,  in  the  southern  hemisphere,  in  the  appearance  and  brightness  of  their 
light  not  unlike  portions  of  the  Milky  Way  of  the  same  apparent  siiie. 
They  are,  generally  speaking,  round,  or  somewhat  oval,  and  the  larger,  which 
deviates  most  from  the  circular  form,  exhibits  the  appearance  of  an  axis 
of  light,  very  ill  defined,  and  by  no  means  strongly  distinguished  from  the 
general  mass,  which  seems  to  open  out  at  its  extremities  into  somewhat  oval 
sweeps,  constituting  the  preceding  and  following  portions  of  its  circumference. 
A  small  patch,  visibly  brighter  than  the  general  light  around,  in  its  follow- 
ing part,  indicates  to  the  naked  eye  the  situation  of  a  very  remarkable 
nebula  (No.  30  Doradiis  of  Bode's  catalogue,)  of  which  more  hereafter. 
The  greater  nubecula  is  situated  between  the  meridians  of  4"  40"  and  6" 
0"  and  the  parallels  of  156°  and  162"  of  N.P.D.,  and  occupies  an  area 
of  about  42  square  degrees.  The  lesser,  between  the  meridians'  0"  28" 
and  l"  15"  and  the  parallels  of  162°  and  165°  N.P.D.  covers  about  ten 
square  degrees.  Their  degree  of  brightness  n^ay  be  judged  of  from  the 
effect  of  strong  moonlight,  which  totally  obliterates  the  lesser,  but  not 
quite  the  greater.  'r.  ;  >     :t 

•  R.A.  20M9-20',  N.P.D.  58°  27'. 

*  It  is  laid  down  nearly  an  hour  wrong  in  ail  the  celestial  charts  and  globe*. 


■ill 


I 


■> 


Il» 


616 


0UTLINE8   OF  ASTRONOMY. 


(893.)  When  examined  through  powerful  telescopeB,  the  constitution 
of  the  nuheculte,  and  especially  of  the  nubecula  major,  is  found  to  be  of 
astonishing  complexity.  The  general  ground  of  both  consists  of  large 
tracts  and  patches  of  nebulosity  in  every  stage  of  resolution,  from  light, 
irresolvable  with  18  inches  of  reflecting  aperture,  up  to  perfectly  separated 
stars  like  the  Milky  Way,  and  clustering  groups  sufficiently  insulated  and 
condensed  to  come  under  the  designation  of  irregular,  and  in  some  cases 
pretty  rich  clusters.  But  besides  those,  there  are  also  nebulae  in  abun- 
dance, both  regular  and  irregular;  globular  clusters  in  every  state  of 
condensation;  and  objects  of  a  nebulous  character  quite  peculiar,  and 
which  have  no  analogue  in  any  other  region  of  the  heavens.  Such  is  the 
concentration  of  these  objects,  that  in  the  area  occupied  by  the  nubecula 
major,  not  fewer  than  278  nebulte  and  clusters  have  been  enumerated, 
besides  50  or  60  outliers,  which  (considering  the  general  barrenness  in 
such  objects  of  the  immediate  neighbourhood)  ought  certainly  to  be 
reckoned  as  its  appendages,  being  about  6}  per  square  degree,  which  very 
far  exceeds  the  average  of  any  other,  even  the  most  crowded  parts  of  the 
nebulous  heavens.  In  the  nubecula  minor,  the  concentration  of  such 
objects  is  less,  though  still  very  striking,  87  having  been  observed  within 
its  area,  and  6  adjacent,  but  outlying.  The  nubeculse,  then,  combine, 
each  within  its  own  area,  characters  which  in  the  rest  of  the  heavens  are 
no  less  strikingly  8eparated,-:-viz.,  those  of  the  galactic  and  the  nebular 
system.  Qlobular  clusters  (except  in  one  region  of  small  extent)  and 
nebulae  of  regular  elliptic  forms  are  comparatively  rare  in  the  Milky  Way, 
and  are  found  congregated  in  the  greatest  abundance  in  a  part  of  the 
heavens,  the  most  remote  possible  from  that  circle ;  whereas,  in  the  nube- 
culae,  they  are  indiscriminately  mixed  with  the  general  starry  ground,  and 
with  irregular  though  small  nebulae. 

(894.)  This  combination  of  characters,  rightly  considered,  is  in  a  high 
degree  instructive,  affording  an  insight  into  the  probable  comparative  dis- 
tance of  stars  and  nebulae,  and  the  real  brightness  of  individual  stars  as 
compared  one  with  another.  Taking  the  apparent  semidianietcr  of  the 
nubecula  major  at  3°,  and  regarding  its  solid  form  as,  roughly  speaking, 
spherical,  its  nearest  and  most  remote  parts  differ  in  their  distance  from 
us  by  a  little  more  than  a  tenth  part  of  our  distance  from  it'  centre.  The 
brightness  of  objects  situated  in  its  nearer  portions,  therefo*e,  cannot  be 
much  exaggerated,  nor  that  of  its  remoter  much  enfeebled,  by  their  differ- 
ence of  distance ;  yet  within  this  globular  space,  we  have  collected  upwards 
of  600  stars  of  the  7th,  8th,  9th,  and  10th  magnitudes,  nearly  300 
nebulae,  and  globular  and  other  clusters,  of  all  degrees  ofresolvbility,  and 
smaller  scattered  stars  innumerable  of  every  inferior  magnitude,  from  the 


10th 
Dcbul 
but  0 
that 
realit 


THE   MAQELLANIO  CLOUDS. 


617 


10th  to  such  as  by  their  nmltitude  and  minuteness  constitute  irresolvable 
nebulosity,  extending  over  tracts  of  many  square  degrees.  Were  there 
but  one  such  object,  it  might  be  maintained  without  utter  improbability 
that  its  apparent  sphericity  is  only  an  effect  of  foreshortening,  and  that  in 
reality  a  much  greater  proportional  difference  of  distance  between  its 
nearer  and  more  remote  parts  exists.  But  such  an  adjustment,  improba- 
ble enough  in  one  case,  must  be  rejected  as  too  much  so  for  fair  argument 
in  two.  It  must,  therefore,  be  taken  as  a  demonstrated  fact,  that  stars  of 
the  7th  or  8th  magnitude  and  irresolvable  nebula  may  co-exist  within 
limitn  of  distance  not  differing  in  proportion  more  than  as  9  to  10,  a  con- 
clusion which  must  inspire  some  degree  of  caution  in  admitting,  as  certain, 
many  of  the  consequences  which  have  been  rather  strongly  dwelt  upon  in 
the  foregoing  pages.  •  .      • 

(895.)  Immediately  preceding  the  centre  of  the  nubecula  minor,  and 
undoubtedly  belonging  to  the  same  group,  occurs  the  superb  globular 
cluster.  No.  47,  Toucani  of  Bode,  very  visible  to  the  naked  eye,  and  one 
of  the  finest  objects  of  this  kind  in  the  heavens.  It  consists  of  a  very 
condensed,  spherical  mr  ss  of  stars,  of  a  pale  rose-colour,  concentrically 
enclosed  in  a  much  less  condensed  globe  of  white  ones,  15'  or  20'  in 
diameter.  This  is  the  first  in  order  of  the  list  of  such  clusters  in 
Art.  867. 

(896.)  Within  the  nubecula  major,  as  already  mentioned,  and  faintly 
visible  to  the  naked  eye,  is  the  singular  nebula  (marked  as  the  star  30 
Doradfis  in  Bode's  Catalogue)  noticed  by  Lacaille  as  resembling  the  nu- 
cleus of  a  small  comet.  It  occupies  about  one-500th  part  of  the  whole 
area  of  the  nubecula,  and  is  so  satisfactorily  represented  in  plate  V.,  fig.  1, 
as  to  render  further  description  superfluous. 

(897.)  We  shall  conclude  this  chapter  by  the  mention  of  two  phseno- 
mena,  which  seem  to  indicate  the  existence  of  some  slight  degree  of  nebu- 
losity about  the  sun  itself,  and  even  to  place  it  in  the  list  of  nebulous 
stars.  The  first  is  that  called  the  zodiacal  light,  which  may  be  seen  any 
very  clear  evening,  soon  after  sunset,  about  the  months  of  March,  April, 
and  May,  or  at  the  opposite  seasons  before  sunrise,  as  a  cone  or  lenticu- 
larly-shaped  light,  extending  from  the  horizon  obliquely  upwards,  and 
following  generally  the  course  of  the  ecliptic,  or  rather  that  of  the  sun's 
equator.  The  apparent  angtilar  distance  of  its  vertex  from  the  sun  varies, 
according  to  circumstances,  from  40"  to  90',  and  the  breadth  of  its  base 
perpendicular  to  its  axis  from  8°  to  SO**.  It  is  extremely  faint  and  ill 
defined,  at  least  in  this  climate,  though  better  seen  in  tropical  regions,  but 
caunot  be  mistaken  for  any  atmospheric  meteor  or  aurora  borealis.  It  is 
manifestly  in  the  nature  of  a  lenticularly-formed  envelope,  surrounding 


wi 


M 


518 


OUTLINES  OP  ASTRONOMY, 


/ 


the  sun.  and  extending  beyond  the  orbits  of  Mercury  and  Venui,  and 
nearly,  perhaps  quite,  attaining  that  of  the  earth,  since  its  vertex  bos  been 
seen  fully  90°  from  the  sun's  place  in  a  groat  circle.  It  may  be  conjec- 
tured to  be  no  other  than  the  denser  part  of  that  medium,  which,  we 
have  some  reason  to  balieve,  resists  the  motion  of  comets ;  loaded,  per- 
haps, with  the  actual  materials  of  the  tails  of  millions  of  those  bodies,  of 
which  they  have  been  stripped  in  their  successive  perihelion  passages 
(Art.  660).  An  atmosphere  of  the  sun,  in  any  proper  sense  of  the  word, 
it  cannot  be,  since  the  existence  of  a  gaseous  envelope  propagating  pres- 
sure from  part  to  part ;  subject  to  mutual  friction  in  its  strata,  and  there- 
fore rotating  in  the  same  or  nearly  the  same  time  with  the  central  body, 
and  of  such  dimensions  and  ellipticity,  is  utterly  incompatible  with  dyna- 
mical laws.  If  its  particles  have  inertia,  they  must  necessarily  stand  with 
respect  to  the  sun  in  the  relation  of  separate  and  independent  minute 
planets,  each  having  its  own  orbit,  plane  of  motion,  and  periodic  time. 
The  total  mass  being  almost  nothing  compared  to  that  of  the  sun,  mutual 
perturbation  is  out  of  the  qu'^stion,  though  collisions  among  such  as  may 
cross  ench  other's  paths  may  operate  in  the  course  of  indefinite  ages  to 
effect  a  subsidence  of  at  least  some  portion  of  it  into  the  body  of  the  sun 
or  those  of  the  planets. 

(808.)  Nothing  prevents  that  these  particles,  or  eome  among  tbem, 
may  have  some  tangible  size,  and  be  at  very  great  distances  from  each 
other.  Compared  with  planets  visible  in  our  most  powerful  telescopes, 
rocks  and  stony  masses  of  great  size  and  weight  would  be  but  as  the  im- 
palpable dust  which  a  sunbeam  renders  visible  as  a  sheet  of  light,  when 
streaming  through  a  narrow  chink  into  a  davk  chamber.  It  is  a  fa(*t, 
established  by  the  most  indisputable  evidence,  that  stony  masses  and 
lunips  of  iron  do  occasionally,  and  indeed  by  no  means  unfrcquently,  fall 
upon  the  earth  from  the  higher  regions  of  our  atmosphere  (where  it  is 
obviously  impossible  they  can  have  been  generated),  and  that  they  have 
done  so  from  the  earliest  times  ^ni  history.  Four  instances  are  recorded 
of  persons  beiu^  killed  by  their  fall.  A  block  of  stone  fell  at  iEgos 
Potamos,  B. C  465,  as  large  as  two  mill-stones;  another  at  Nami,  in  921, 
projected,  like  a  rtx^k.  four  feet  above  the  surface  of  the  river,  into  which 
it  was  seea  to  fkll.  The  emperor  Jehangire  had  a  sword  forged  from  a 
mas«  of  meteoric  iron  which  fell,  in  1620,  at  Jahlinder,  in  the  Punjab.' 
Sixttten  instances  of  the  fall  of  stones  in  the  British  Isles  are  well  authi  u- 
ticated  to  have  occurr  i  since  1620,  one  of  them  in  Loni^n.  In  1803, 
on  the  26th  of  April,  thousands  of  stones  were  scattered  by  the  explosion 

*■  Hee  the  emperor's  own  vciry  remarkable  account  of  the  occurrence,  translated  in 
Trans.  1793,  p.  202. 


METEOROLITES   AND   SIIOOTINO   STARS. 


610 


into  fragmonts  of  a  large  fiery  globo  over  a  region  of  twenty  or  tbirty 
square  miles  around  the  town  of  L'Aiglc,  in  Normundy.  The  fact  occurred 
at  mid-day,  and  the  circumBtanoofl  were  oiTiciully  verified  by  a  comniisHion 
of  the  French  government.'  These,  and  innumerable  other  instances,' 
fully  establish  the  general  fact ;  and  after  vain  attempts  to  account  for  it 
by  volcanic  projection,  either  from  the  earth  or  the  moon,  the  planetary 
nature  of  these  bodies  seems  at  length  to  be  almost  generally  admitted. 
The  heat  which  they  possess  when  fallen,  the  igneous  phaonomcna  which 
accompany  them,  their  explosion  on  arriving  within  the  denser  regions  of 
our  atmosphere,  &c.,  are  all  sufficiently  accounted  for  on  physical  priuci- 
pies,  by  the  condensation  of  the  air  before  them  in  consequence  of  their 
enormous  velocity,  and  by  the  relations  of  air  in  a  highly  attenuated  state 
to  heat.' 

(899.)  Besides  stony  and  metallic  masses,  however,  it  is  probable  that 
bodies  of  very  different  natures,  or  at  least  states  of  aggi-egation,  are  thus 
circulating  round  the  sun.  Shooting  stars,  often  followed  by  long  trains 
of  light,  and  those  great  fiery  globes,  of  more  rare,  but  not  very  uncommon 
occurrence,  whicli  are  seen  traversing  the  upper  regions  of  our  atmosphere, 
sometimes  leaving  trains  behind  them,  remaining  for  many  minutes,  some- 
times bursting  with  a  loud  explosion,  sometimes  becoming  quietly  extinct, 
may  not  unreasonably  be  presumed  to  be  bodies  extraneous  to  our  planet, 
which  only  become  visible  when  in  the  act  of  grazing  the  confines  of  our 
atmosphere.  Among  the  last  mentioned  meteors  are  some  which  can 
hardly  be  supposed  solid  masses.  The  remarkable  meteor  of  Aug.  18, 
1783,  travel  ^ni  the  whole  of  Europe,  from  Shetland  to  Rome,  with  a 
velocity  f  about  30  miles  per  second,  at  a  height  of  50  miles  from  the 
surface  i^*  tin,  earth,  with  a  light  greit'.y  surpassing  that  of  the  full  moon, 
and  a  iwal  diameter  of  fully  half  a  mile.  Yet  with  these  vast  dimensions, 
it  ciaanged  its  form  visibly,  and  at  length  quietly  separated  into  several 
di*tinct  bodies,  accompanying  each  other  in  parallel  courses,  and  each  fol- 
low«^l  by  a  tail  or  train. 

(900.)  There  are  circumstances  in  the  history  of  shooting  stars,  which 
very  strongly  corroborate  the  idea  of  their  extraneous  or  cosmical  origin, 
and  their  circulation  round  the  sun  in  definite  orbits.  On  several  occa- 
sioDB  they  have  been  observed  to  appear  in  unusual,  and,  indeed,  astonish 


'M 


'  See  M.  Biot's  report  in  M^m.  de  I'lnstitut.  1806. 

'  See  a  list  of  upwards  of  200,  published  by  Chladni,  Annates  du  Bureau  des  Lon 
gitudes  de  France,  1825. 

'  Edinburgh  Review,  Jan.  1848,  p.  195.  It  is  very  remarkable  that  no  new  chemical 
element  has  been  detected  in  any  of  the  numerous  meteorolites  which  have  been  sub- 
jected to  analysis. 


520 


OUTLINES  OF  ASTRONOMT. 


ing  numbers,  so  as  to  convey  the  idea  of  a  shower  of  rockets,  or  of  snow- 
flakes  falling,  and  brilliantly  illuminating  the  whole  heavens  for  hours 
together,  and  that  not  in  one  locality,  but  over  whole  continents  and 
oceans,  and  even  (in  one  instance)  in  both  hemispheres.  Now  it  is  ex- 
tremely remarkable  that,  whenever  this  great  display  has  been  exhibited 
(at  least  in  modem  times),  it  has  uniformly  happened  on  the  night  be- 
tween the  12th  and  18th,  or  on  that  between  the  13th  and  14th  of  No- 
vember. Such  cases  occurred  in  1799,  1823,  1832,  1833,  and  1834. 
On  tracing  back  the  records  of  similar  phaenomena,  it  has  been  ascertained, 
moreover,  that  more  often  those  identical  nights,  but  sometimes  those 
immediately  adjacent,  have  been,  time  out  of  mind,  habitually  signalized 
by  such  exhibitions.  Another  annually  recurring  epoch,  in  which,  though 
far  less  brilliant,  the  display  of  meteors  is  more  certain  (for  that  of  No- 
vember is  often  interrupted  for  a  great  many  years),  is  that  of  the  10th 
of  August,  on  which  night,  and  on  the  9th  2>nd  11th,  numerous,  large, 
and  bright  shooting  stars,  with  trains,  are  almost  sure  to  be  seen.  Other 
epochs  of  periodic  recurrence,  less  marked  than  the  above,  have  also  been 
to  a  certain  ext«nt  established. 

(901.)  It  is  impossible  to  attribute  such  a  recurrence  of  identical  dates 
of  very  remarkable  phaBiiomena  to  accident.  Annual  periodicity,  irre- 
spective of  geographical  position,  refers  us  at  once  to  the  place  ocupied  by 
the  earth  in  its  annual  orbit,  and  leads  direct  to  the  conclusion  that  at  that 
place  the  earth  incurs  a  liability  to  frequent  encounters  or  concurrences 
with  a  stream  of  meteors  in  their  progress  of  circulation  round  the  sun. 
Let  us  test  this  idea  by  pursuing  it  into  some  of  its  consequences.  In 
the  first  places  then,  supposing  the  earth  to  plunge,  in  its  yearly  circuit, 
into  a  uniform  ring  of  innumerable  small  meteor-planets,  of  such  breadth 
as  would  be  traversed  by  it  in  one  or  two  days ;  since  during  this  small 
time  the  motions,  whether  of  the  earth  or  of  each  individual  meteor,  may 
be  taken  as  uniform  and  rectilinear,  and  those  of  all  the  latter  (at  the 
place  and  time)  parallel,  or  very  nearly  so,  it  will  follow  that  the  relative 
motion  of  the  meteors  referred  to  the  earth  as  at  rest,  will  be  also  uniform, 
rectilinear,  and  parallel.  Viewed,  therefore,  from  the  centre  of  the  earth 
(or  from  any  point  in  its  circumference,  if  we  neglect  the  diurnal  velocity 
as  very  small  compared  with  the  annual)  they  will  all  appear  to  diverge 
from  a  common  poiut,  fixed  in  relation  to  the  celestial  sphere,  as  if  ema- 
nating from  a  sidereal  apex  (Art.  115). 

(902.)  Now  this  is  precisely  what  actually  happens.  The  meteors  of 
the  12th — 14th  of  November,  or  at  least  the  vast  majority  of  them,  de- 
scribe apparently  arcs  of  great  circles,  passing  through  or  near  y  Leonis. 
No  matter  what  the  situation  of  that  star  with  respect  to  the  horizon  or 


PERIODICAL  APPEARANCE   OF  METEORS. 


521 


to  its  east  and  west  points  may  be  at  the  time  of  observation,  the  paths 
of  the  meteors  all  appear  to  diverge  from  that  star.  On  the  9  th — 11th 
of  August  the  geometrical  fact  is  the  same,  the  apex  only  differing;  B 
Camelopardali  being  for  that  epoch  the  point  of  divergence.  As  we  need 
not  suppose  the  meteoric  ring  coincident  in  its  plane  with  the  ecliptic, 
and  as  for  a  ring  of  meteors  we  may  substitute  an  elliptic  annulus  of  any 
reasonable  ezcentricity,  so  that  both  the  velocity  and  direction  of  each 
meteor  may  differ  to  any  extent  from  the  earth's,  there  is  nothing  in  the 
great  and  obvious  difference  in  latitude  of  these  apices  at  all  militating 
against  the  conclusion. 

(903.)  If  the  meteors  be  uniformly  distributed  in  such  a  ring  or  ellip- 
tic annulus,  the  earth's  encounter  with  them  in  every  revolution  will  be 
certain,  if  it  occur  once.  But  if  the  ring  be  broken,  if  it  be  a  succession 
of  groups  revolving  in  an  ellipse  in  a  period  not  identical  with  that  of  the 
earth,  years  may  pass  without  a  rencontre ;  and  when  such  happen,  they 
may  differ  to  any  extent  in  their  intensity  of  character,  according  as  richer 
or  poorer  groups  have  been  encountered. 

(904.)  No  other  plausible  explanation  of  these  highly  characteristic 
features  (the  annual  periodicity,  and  divergence  from  a  common  apex, 
always  alike /or  each  respective  epoch)  have  been  even  attempted,  and  ac- 
cordingly the  opinion  is  generally  gaining  ground  among  astronomers  that 
shooting  stars  belong  to  their  department  of  science,  and  great  interest  is 
excited  in  their  observation  and  the  further  development  of  their  laws. 
The  most  connected  and  systematic  series  of  observations  of  them,  having 
for  their  object  to  trace  out  their  relative  paths  with  respect  to  the  earth, 
are  those  of  Benzenberg  and  Brandes,  who,  by  noting  the  instants  and 
apparent  places  of  appearance  and  extinction,  as  well  as  the  precise  appa- 
rent paths  among  the  stars,  of  individual  meteors,  from  the  extremities 
of  a  measured  base  line  nearly  50,000  feet  in  length,  were  led  to  con- 
clude that  their  heights  at  the  instant  of  their  appearance  and  disappear- 
ance vary  from  16  miles  to  140,  and  their  relative  velocities  from  18  to 
86  miles  per  second,  velocities  so  great  as  clearly  to  indicate  an  indepen- 
dent planetary  circulation  round  the  sun.  [A  very  remarkable  meteor 
or  bolide,  which  appeared  on  the  19th  August,  1847,  was  observed  at 
Dieppe  and  at  Paris,  with  suScient  precision  to  admit  of  calculation  of  the 
elements  of  its  orbit  in  absolute  space.  This  calculation  has  been  per- 
formed by  M.  Petit,  director  of  the  observatory  of  Toulouse,  and  the  fol- 
lowing hyperbolic  elements  of  its  orbit  round  the  sun  are  stated  by  him 
(Astr.  Nachr.  701)  as  its  result;  viz.,  Semimajor  axis  =  — 0-3240083; 
excentricity  =  3-95130;  perihelion  distance  =  0-95626 ;  inclination  to 
plane  of  the  earth's  equator,  18"  20'  18" ;  ascending  node  on  the  samo 


1 

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j 
1 

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im 

i 

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ill 

r' 

m 

f 

ml 

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B 

n^i 

H! 


i'o 


hi 


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i 


522 


OUTLINES   OF  ASTRONOMY. 


plane,  10*^  84'  48";  motion  direct.  According  to  this  calculation,  the 
body  would  have  occupied  no  less  than  87340  years  in  travelling  from  the 
distance  of  the  nearest  fixed  star  supposed  to  have  a  parallax  of  1".] 

(905.)  It  is  by  no  means  inconceivable  that  the  earth  approaching  to 
such  as  differ  but  little  from  it  in  direction  and  velocity,  may  have  at- 
tached many  of  them  to  it  as  permanent  satellites,  and  of  these  there  may 
be  Bonje  so  large,  and  of  such  texture  and  solidity,  as  to  shine  by  reflected 
light,  and  become  visible  (such,  at  least,  as  are  very  near  the  earth)  for  a 
brief  moment,  suffering  extinction  by  plunging  into  the  earth's  shadow ; 
in  other  words  undergoing  total  eclipse.  Sir  John  Lubbock  is  of  opinion 
that  such  is  the  case,  and  has  given  geometrical  formulas  for  calculating 
their  distances  from  observations  of  this  nature.'  The  observations  of  M. 
Petit  would  lead  us  to  believe  in  the  existence  of  at  least  one  such  body, 
revolving  round  the  earth,  as  a  satellite,  in  about  8  hours  20  minutes,  and 
therefore  at  a  distance  equal  to  2-513  radii  of  the  earth  from  its  centre, 


or  5000  miles  above  its  surface.' 


\ 


<  Phil.  Mag.  Lond.  Ed.  Dub.  1848,  p.  80.    ■ 

*  Comptes  Rendus,  Oct.  12,  1846,  and  Aug.  9,  1847. 


nAIVRAL  UNITS  OF  TIME. 


528 


m 


PART  IV. 


OP    THE    ACCOUNT    OP    TIME. 


CHAPTER  XVIII. 

NATURAL  UNITS  OP  TIME. — RELATION  OP  THE  SIDEREAL  TO  THE  SOLAR 
DAY  AFFECTED  BY  PRECESSION. — INCOMMENSURABILITY  OP  THE  DAY 
AND  YEAR. —  ITS  INCONVENIENCE. —  HOW  OBVIATED. — THE  JULIAN 
CALENDAR.  —  IRREGULARITIES  AT  ITS  FIRST  INTRODUCTION. — RE- 
FORMED BY  AUGUSTUS. — GREGORIAN  REFORMATION.  —  SOLAR  AND 
LUNAR  CYCLES.  —  INDICTION. — JULIAN  PERIOD.  —  TABLE  OF  CHRO- 
NOLOGICAL ERAS.  —  RULES  FOR  CALCULATING  THE  DAYS  ELAPSED 
BETWEEN   GIVEN   DATES.  —  EQUINOCTIAL  TIME. 

(906.)  Time,  like  distance,  may  be  measured  by  comparison  with  stan- 
dards of  any  length,  and  all  that  is  requisite  for  ascertaining  correctly  the 
length  of  any  interval,  is  to  be  able  to  apply  the  standard  to  the  interval 
throughout  its  whole  extent,  without  overlapping  on  the  one  hand,  or 
leaving  unmeasured  vacancies  on  the  other;  to  determine,  without  the 
possible  error  of  a  unit,  the  number  of  integer  standards  which  the  inter- 
val admits  of  being  interposed  between  its  beginning  and  end ;  and  to 
estimate  precisely  the  fraction,  over  and  above  an  integer,  which  remains 
when  all  the  possible  integers  are  subtracted. 

(907.)  But  though  all  standard  units  of  time  are  equally  possible,  the- 
oretically speaking,  yet  all  are  not,  practically,  equally  convenient.  The 
solar  day  is  a  natural  interval  which  the  wants  and  occupations  of  man  in 
every  state  of  society  force  upon  him,  and  compel  him  to  adopt  as  his 
fundamental  unit  of  time.  Its  length  as  estimated  from  the  departure 
of  the  sun  from  a  given  meridian,  and  its  next  return  to  the  same,  is  sub- 
ject, it  is  true,  to  an  annual  fluctuation  in  excess  and  defect  of  its  mean 
value,  amounting  at  its  maximum  to  full  haK  a  minute.  But  except  for 
astronomical  purposes,  this  is  too  small  a  change  to  interfere  in  the  slight- 
est degree  with  its  use,  or  to  attract  any  attention,  and  the  tacit  substita* 


m 

[ij,;, 

41 


''^ 


i  ■■ 


Jli 


m\ 


lit 
fill 

fi 


524 


OUTLINES  OF  i;STBONOMT. 


tion  of  its  mean  for  its  true  (or  variable)  value  may  be  considered  as 
having  been  made  from  tbe  earliest  ages,  by  the  ignorance  of  mankind 
that  any  such  fluctuation  existed. 

(908.)  The  time  occupied  by  one  complete  rotation  of  the  earth  on  its 
axis,  or  the  mean'  sidereal  day,  may  be  shown,  on  dynamical  principles,  to 
be  subject  to  no  variation  from  any  external  cause,  and  although  its  dura- 
tion would  be  shortened  by  contraction  *a  the  dimensions  of  the  globe 
itself,  such  as  might  arise  from  the  gradual  escape  of  \ts  internal  heat, 
and  consequent  refrigeration  and  shrinking  of  the  wholfa  mass,  yet  theory, 
on  the  one  hand,  has  rendered  it  almost  certain  that  this  cause  cannot  have 
eflfected  any  perceptible  amount  of  change  during  the  history  of  the  huma  i 
race ;  and,  on  the  other,  the  comparison  of  ancient  and  modern  observa- 
tions affords  every  corroboration  to  this  conclusion.  From  such  compari- 
sons, Laplace  has  concluded  that  the  sidereal  day  has  not  changed  by  so 
much  as  one  hundredth  of  a  second  since  the  time  of  Hipparchus.  The 
mean  sidereal  day  therefore  possesses  in  perfection  the  essential  quality  of 
a  standard  unit,  that  of  complete  invariahility.  The  same  is  true  of  the 
mean  sidereal  year,  if  estimated  upon  an  average  sufficiently  large  to  com- 
pensate the  minute  fluctuations  arising  from  the  periodical  variations  of  the 
major  axis  of  the  earth's  orbit  due  to  planetary  perturbation  (Art.  668.) 

(909.)  The  mean  solar  day  is  an  immediato  derivative  of  the  sidereal 
day  and  year,  being  connected  with  them  by  the  same  relation  which  de- 
termines the  synodic  from  the  sidereal  revolutions  of  any  two  planets  or 
other  revolving  bodies  (Art.  418.)  The  ex(ict  determination  of  the  ratio 
of  the  sidereal  to  the  solar  day,  which  is  a  point  of  the  utmost  importance 
in  astronomy,  is  however,  in  some  degree,  complicated  by  tl  i  effect  of 
precession,  which  renders  it  necessary  to  distinguish  between  the  absolute 
time  of  the  earth's  rotation  on  its  axis,  (the  real  natural  and  invaiiable 
standard  of  comparison,)  and  the  mean  interval  between  two  successive 
returns  of  a  given  star  to  the  same  meridian,  or  rather  of  a  given 
meridian  to  the  same  star,  which  not  only  differs  by  a  minute  quantity 
from  the  sidereal  day,  but  is  actually  not  the  same  for  all  stars.  As 
this  is  a  point  to  which  a  little  difficulty  of  conception  is  apt  to  attach, 
it  will  be  necessary  to  explain  it  in  some  detail.  Suppose  then  it 
the  pole  of  the  ecliptic,  and  P  that  of  the  equinoctial,  A  B  C  D  the  sol- 
stitial and  equinoctial  colures  at  any  given  epoch,  and  V  pqr  the  small 
circle  described  by  P  about  h  in  one  revolution  of  the  equinoxes,  i.  c.  in 
25870  years,  or  9448300  solar  days,  all  projected  on  the  plane  of  the 

*  The  true  sidereal  day  is  variable  by  the  effect  of  nutation ;  but  the  variation  (an 
excessively  minute  fraction  a*'  the  whole)  compensates  itself  in  a  revolution  of  the 
moon's  nodes. 


NATURAL  UNITS  OF  TIME. 


525 


ecliptic  ABC D.  Let  S  be  a  star  anywhere  situated  on  the  ecliptic,  or 
heticeen  it  and  the  small  circle  Vqr.  Then  if  the  pole  P  were  at  rest,  a 
merdian  of  the  earth  setting  out  from  P  S  C,  and  revolving  in  the  direc- 
tion C  B,  will  come  again  to  the  star  after  the  exact  lapse  of  one  sidereal 
day,  or  one  rotation  of  the  earth  on  its  axis.  But  P  is  not  at  rest.  After 
the  lapse  of  one  such  day  it  will  have  come  into  the  situation  (suppose) 
p,  the  vernal  equinox  B  having  retreated  to  b,  and  the  colure  P  C  having 
taken  up  the  new  position  p  c.  Now  a  conical  movement  impressed  on 
the  axis  of  rotation  of  a  globe  already  rotating  is  equivalent  to  a  rotation 
impressed  on  the  whole  globe  round  the  axis  of  the  cone,  in  addition  to 
that  which  the  globe  has  and  retaius  round  its  own  independent  axis  of 
revolution.  Such  a  new  rotation,  in  transferring  P  top,  being  performed 
round  an  axis  passlag  through  n,  will  not  alter  the  situation  of  that  point  \ 
of  the  globe  which  has  it  in  its  zenith.  Hence  it  follows  that  pitc  pass- 
ing  through  it  will  be  the  position  taken  up  by  the  meridian  P  »t  C  after 
the  lapse  of  an  exact  sidereal  day.  But  this  does  not  pass  through  S,  but 
falls  short  of  it  by  the  hour-angle  rt  p  S,  which  is  yet  to  be  described 
before  the  meridian  comes  up  to  the  star.  The  meridian,  then,  has  lost 
so  much  on,  or  lagged  so  much  behind,  the  star  in  the  lapse  of  that  in- 
terval. The  same  is  true  whatever  be  the  arc  Pjp.  After  the  lapse  of 
any  number  of  days,  the  pole  being  transferred  to  p,  the  spherical  angle 
rtp  S  will  measure  the  total  hour-angle  which  the  meridian  has  lost  on  the 
star.  Now  where  8  lies  any  where  between  C  and  r,  this  angle  con- 
tinually increase'i  (though  not  uniformly),  attaining  180°  when  p  comes 
to  r,  and  still  (as  will  appear  by  following  out  the  movement  beyond  r) 
increasing  thence  till  it  attains  860^  when  p  has  completed  its  circuit 


i;4 


»i;'i 


■if 


11: 


I  1     I 


;  ! 


i 


'Imi 


526 


OUTLINES   OF   ASTRONOMT. 


Thus  in  a  whole  revolution  of  the  equinoxes,  the  meridian  will  have  lost 
one  exact  revolution  upon  the  star,  or  in  9448300  sidereal  days,  will  have 
re-attained  the  star  only  9448299  times :  in  other  words,  the  length  of  the 
day  measured  by  the  mean  of  the  successive  arrivals  of  any  star  outside 
of  the  circle  Vp  qr  on  one  and  the  same  meridian  is  to  the  absolute  time 
of  rotation  of  the  earth  on  its  axis  as  9448300 :  9448299,  or  as  1-00000011 


(010.)  It  is  otherwise  of  a  star  situated  toithin  this  circle,  as  at  cr.  For 
euch  a  star  the  angle  ftpo,  expressing  the  lagging  of  the  meridian,  in- 
creases to  a  maximum  for  some  situation  of  p  between  q  and  r,  and 
decreases  again  to  o  at  r;  after  which  it  takes  an  opposite  direction,  uud 
the  meridian  begins  to  get  in  advance  of  the  star,  and  continues  to  get 
more  and  more  so,  till  p  has  attained  some  point  between  s  and  P,  where 
the  advance  is  a  maximum,  and  thence  decreases  again  to  o  when  p  has 
completed  its  circuit.  For  any  star  so  situated,  then,  the  mean  of  all  the 
days  so  estimated  through  a  whole  period  of  the  equinoxes  is  an  absolute 
sidereal  day,  as  if  precession  had  no  existence. 

(911.)  If  we  compare  the  sun  with  a  star  situated  in  the  ecliptic,  the 
sidereal  year  is  the  mean  of  all  the  intervals  of  its  arrival  at  that  star 
throughout  indefinite  ages,  or  (without  fear  of  sensible  error^  throughout 
recorded  history.  Now,  if  we  would  calculate  the  synodic  o.  ereal  revo- 
lution of  the  sun  and  of  a  meridian  of  the  earth  by  reference  to  a  star  so 
situated,  according  to  the  principles  of  Art.  418,  we  must  proceed  as  fol- 
lows: Let  D  be  the  length  of  the  mean  solar  day  (or  synodic  day  iff 
question)  d  the  mean  sidereal  revolution  of  the  meridian  with  reference 
to  the  same  star,  and  y  the  sidereal  year.     Then  the  arcs  described  by  the 

sun  and  the  meridian  in  the  interval  D  will  be  respectively  360°  —  and 


860= 


D 


y 


-j.     And  since  the  latter  of  these  exceeds  the  former  by  precisely 


360°,  we  have 


360°^  =  360°-  +  360^ 
d  y 


whence  it  follows  that 

^=  1  +  5=1.00273780, 
d  y 

taking  the  value  of  the  sidereal  year  y  as  given  in  Art.  383,  viz.  365*  G* 

9"  9-6».    But,  as  we  have  seen,  d  is  not  the  absolute  sidereal  day,  but 

exceeds  it  in  the  ratio  1-00000011 : 1.     Hence  to  get  the  value  of  the 

mean  solar  as  expressed  in  absolute  sidereal  days,  the  number  above  set 

down  must  be  increased  in  the  same  ratio,  which  brings  it  to  1  00273791, 


NATURAL  r)SlTB  OF  TIME. 


627 


which  is  the  ratio  of  the  solar  to  the  sidereal  day  actually  in  use  among 
astronomers. 

(912.)  It  would  be  well  for  chronology  if  mankind  would,  or  could 
have  contented  themselves  with  this  one  invariable,  natural,  and  conve- 
nient standard  in  their  reckoning  of  time.  The  ancient  Egyptians  did 
so,  and  by  their  adoption  of  &n  historical  and  official  year  of  365  days 
have  afforded  the  ocly  example  of  a  practical  chronology,  free  from  all 
obscurity  or  complication.  But  the  return  of  the  seasons,  on  which  de- 
pend all  the  more  important  arrangements  and  business  of  cultivated  life, 
is  not  conformabe  to  such  a  multiple  of  the  diurnal  unit.  Their  return 
is  regulated  by  the  tropical  year,  or  the  interval  between  two  successive 
arrivals  of  the  sun  at  the  vernal  equinox,  which,  as  we  have  seen  (Art. 
383),  differs  from  the  sidereal  year  by  reason  of  the  motion  of  the  equi- 
noctial points.  Now  this  motion  is  not  absolutely  uniform,  because  the 
ecliptic,  upon  which  it  is  estimated,  is  gradually,  though  very  slowly, 
changing  its  situation  in  space  under  the  disturbing  influence  of  the  planets 
(Art,  640.)  And  thus  arises  a  variation  in  the  tropical  year,  which  is  de- 
pendent on  the  place  of  the  equinox  (Art.  383.)  The  tropical  year  is 
actually  about  4-21*  shorter  than  it  was  in  the  time  of  Hipparchus.  This 
absence  of  the  most  essential  requisite  for  a  standard,  viz.  invariability, 
renders  it  necessary,  since  we  cannot  help  employing  the  tropical  year  in 
our  reckoning  of  time,  to  adopt  an  arbitrary  or  artificial  value  for  it,  so 
near  the  truth,  as  not  to  admit  of  the  accumulation  of  its  error  for  several 
centuries  producing  any  practical  mischief,  and  thus  satisfying  the  ordi- 
nary wants  of  civil  life ;  while,  for  scientific  purposes,  the  t  epical  year, 
so  adopted,  is  considered  only  as  the  representative  of  a  certain  number 
of  integer  days  and  a  fraction — the  day  being,  in  efiect,  the  only  standard 
employed.  The  case  is  nearly  analogous  to  the  reckoning  of  value  by 
guineas  and  shillings,  an  artificial  relation  of  the  two  coins  being  fixed  by 
law,  near  to,  but  pcarcely  ever  exactly  coincident  with,  the  natural  one, 
determined  by  the  relative  market  price  of  gold  and  silver,  of  which 
either  the  one  or  the  other  —  whichever  is  really  the  most  invariable,  or 
tbe  most  in  use  with  other  nations,  —  may  be  assumed  as  the  true  theo- 
retical standard  of  value. 

(913.)  The  other  inconvenience  of  the  tropical  year  as  a  greater  unit 
is  its  incommensurability  with  the  lesser.  In  our  measure  of  space  all 
our  subdivisions  are  into  aliquot  parts :  a  yard  is  three  feet,  a  mile  eight 
furlongs,  &c.  But  a  year  is  no  exact  number  of  days,  nor  an  integer 
number  with  any  exact  fraction,  as  one  third  or  one  fourth,  over  and  above ; 
but  the  surplus  is  an  incommensurable  fraction,  composed  of  hours, 
niinutes;  seconds,  &c.,  which  produces  the  same  kind  of  inconvenience  in 


in  I 


■'"]! 


'i 


I' 


,1,; 


m 


« 


i 


528 


OUTLINES   OF  ASTRONOMT. 


in  the  reckoning  of  time  that  it  would  do  in  that  of  money,  if  we  had 
gold  coins  of  the  value  of  twenty-one  shillings,  with  odd  pence  and  far- 
things, and  a  fraction  of  a  farthing  over.  For  this,  however,  there  is  no 
remedy  but  to  keep  a  strict  register  of  the  surplus  fractions ;  and,  when 
they  amount  to  a  whole  day,  oast  them  over  into  the  integer  account. 

(914.)  To  do  this  in  the  simplest  and  most  convenient  manner  is  the 
object  of  a  well-adjusted  calendar.  In  the  Gregorian  calendar,  which  we 
follow,  it  is  accomplished  with  as  much  simplicity  and  neatness  as  the 
case  admits,  by  carrying  a  little  farther  than  is  done  above,  the  principle 
of  an  assumed  or  artificial  year,  and  adopting  two  such  years,  both  con- 
sisting of  an  exact  integer  number  of  days,  viz.  one  of  365  and  the  other 
of  866,  and  laying  down  a  simple  and  easily  remembered  rule  for  the 
order  in  which  these  yeart^  shall  succeed  each  other  in  the  civil  reckoning 
of  time,  so  that  during  the  lapse  of  at  least  some  thousands  of  years  the 
sum  of  the  integer  artificial,  or  Gregorian,  years  elapsed  shall  not  differ 
from  the  same  number  of  real  tropical  years  by  a  whole  day.  By  this 
contrivance,  the  equinoxes  and  solstices  will  always  fall  on  days  similarly 
situated,  and  bearing  the  same  name  in  each  Gregorian  year;  and  the 
seasons  will  for  ever  correspond  to  the  same  months,  instead  of  running 
the  round  of  the  whole  year,  as  they  must  do  upon  any  other  system  of 
reckoning,  and  used,  in  fact,  to  do  before  this  was  adopted  as  a  matter  of 
ignorant  haphazard  in  the  Greek  and  ^oman  chronology,  and  of  strictly 
defined  and  superstitiously  rigorous  observance  in  the  Egyptian. 

(915.)  The  Gregorian  rule  is  as  follows  '• — The  years  are  denominated 
as  years  current  (not  as  years  elapsed)  from  the  midnight  between  the 
Slst  of  December  and  the  1st  of  January  immediately  subsequent  to  the 
birth  of  Christ,  according  to  the  chronological  determination  of  that  event 
by  DionysiuT  Exiguus.  Every  year  whose  number  is  not  divisible  by  4 
without  remainder,  consists  of  365  days ;  every  year  which  is  so  divisible, 
but  is  not  divisible  by  100,  of  366 ;  every  year  divisible  by  100,  but  not 
by  400,  again  of  365  j  and  every  year  divisible  by  400,  again  of  366. 
For  example,  the  year  1833  not  being  divisible  by  4,  consists  of  365 
days;  1836  of  366;  1800  and  1900  of  365  each;  but  2000  of  366.  In 
order  to  see  how  near  this  rule  will  bring  us  to  the  truth,  let  us  see  what 
number  of  days  10000  Gregorian  years  will  contain,  beginning  with  the 
year  a.  D.  1.  Now,  in  10000,  the  numbers  not  divisible  by  4  will  be  f  of 
10000  or  7500 ;  those  divisible  by  100,  but  not  by  400,  will  in  like  manner 
be  f  of  100,  or  75 ;  so  that,  in  the  10000  years  in  question,  7575  consist 
of  366,  and  the  remaining  2425  of  365,  producing  in  all  3652425  days, 
which  would  give  for  an  average  of  each  year,  one  with  another,  365**2425. 
The  actual  value  of  the  tropical  year,  (art.  388)  reduced  into  a  decimal 


NATURAL  UNITS  OF  TIME. 


629 


fractbn,  is  365-24224,  so  the  error  in  the  Gregorian  rale  on  10000  of 
the  present  tropical  years,  is  2  6,  or  2*  14"  24" ;  that  is  to  say,  less  than 
a  day  in  8000  years ;  which  is  more  than  sufficient  for  all  human  purposes, 
those  of  the  astronomer  excepted,  who  is  in  no  danger  of  being  led  into 
error  from  this  cause.     Even  this  error  is  avoided  by  extending  the 
wording  of  the  Gregorian  rule  one  step  farther  than  its  contrivers  probably 
thought  it  worth  while  to  go,  and  declaring  that  vears  divisible  by  4000 
should  consist  of  365  days.     This  would  take  off  two  integer  days  from 
the  above  calculated  number,  and  2'5  from  a  larger  average;  making  the 
sum  of  days  in  100000  Gregorian  years,  36524225,  which  differs  only  by 
a  single  day  from  100000  real  tropical  years,  such  as  they  exist  at  present. 
(916.)  In  the  historical  dating  of  events  there  is  no  year  A.  d.  0.    The 
year  immediately  previous  to  A.  D.  1,  is  always  called  b.  o.  1.    This  must 
always  be  borne  in  mind  in  reckoning  chronological  and  astronomical 
intervals.     The  sum  of  the  nominal  years  b.  0.  and  a.  d.  must  be  dimin- 
ished by  1.    Thus,  from  Jan.  1,  b.  o.  4718,  to  Jan.  1,  1582,  the  years 
elapsed  are  not  6295,  but  6294. 

(917.)  As  any  distance  along  a  high  road  might,  though  in  a  rather 
inconvenient  and  roundabout  way,  be  expressed  without  introducing  error 
by  setting  up  a  series  of  milestones,  at  intervals  of  unequal  lengths,  so 
that  every  fourth  mile,  fbr  instance,  should  be  a  yard  longer  than  the  rest, 
or  according  to  any  other  fixed  rule  j  taking  care  only  to  mark  the  stones 
80  as  to  leave  room  for  no  mistake,  and  to  advertise  all  travellers  of  the 
difference  of  lengths  and  their  order  of  succession ;  so  may  any  interval 
of  time  be  expressed  correctly  by  stating  in  what  Gregorian  years  it  begins 
and  ends,  and  wJiereabouts  in  each.  For  this  statement  coupled  with  the 
declaratory  rule,  enables  us  to  say  how  many  integer  years  are  to  be 
reckoned  at  365,  and  how  many  at  366  days.  The  latter  years  are  called 
bissextiles,  or  leap-years,  and  the  surplus  days  thus  thrown  into  the 
reckoning  are  called  intercalary  or  leap  days. 

(918.)  If  the  Gregorian  rule,  as  above  stated,  had  always  and  in  all 
countries  been  known  and  followed,  nothing  would  be  easier  than  to  reckon 
the  number  of  days  elapsed  between  the  present  time,  and  any  historical 
recorded  event.  But  this  is  not  the  case ;  and  the  history  of  the  calendar, 
with  reference  to  chronology,  or  to  the  calculation  of  ancient  observations, 
maj  be  compared  to  that  of  a  clock,  going  regularly  when  left  to  itself, 
but  sometimes  forgotten  to  be  wound  up ;  and  when  wound,  sometimes 
set  forward,  sometimes  backward,  either  to  serve  particular  purposes  and 
private  interests,  or  to  rectify  blunders  in  setting.  Such,  at  least,  appears 
to  have  been  the  case  with  the  Roman  calendar,  in  which  our  own  origi- 
nates, from  the  time  of  Numa  to  that  of  Julius  Caesar,  when  the  lunar 
34 


'jii! 


i .  n 


m 


I ' 


if*!  I  I 


m 


!  I 


r^^ 


ill! 


530 


OUTLINES  OF  ASTRONOMY. 


year  of  13  months,  or  855  days,  was  augmented  at  pleasaro  to  correspond 
to  the  solar,  by  which  the  seasons  are  determined,  by  the  arbitrary  inter- 
calations of  the  priests,  and  the  usurpations  of  the  decemvirs  and  other 
magistrates,  till  the  confusion  became  inextricable.  To  Julius  Ca3sar, 
assisted  by  Sosigenes,  an  eminent  Alexandrian  astronomer  and  mathema- 
tician, we  owe  the  neat  contrivance  of  the  two  years  of  865  and  366  days, 
and  the  insertion  of  one  bissextile  aftev  three  common  years.  This  im- 
portant change  took  place  in  the  45th  year  before  Christ,  which  he  ordered 
to  couirnenco  on  the  Ist  of  January,  bein(/  the  day  of  the  new  moon  i.n- 
mcdiatdi/  following  the  winter  solstice  of  the  year  before.  We  may  judge 
of  the  state  into  which  the  reckoning  of  time  had  fallen,  by  the  fact,  that 
to  introduce  the  new  system  it  was  necessary  to  enact  that  the  previous 
year,  46  B.  c,  should  consist  of  445  days,  a  circumstance  which  obtained 
for  it  the  epithet  of  "  the  year  of  confusion." 

(019.)  Had  Caesar  lived  to  carry  out  into  practical  effect,  as  Chief 
Pontiff,  his  own  reformation,  an  inconvenience  would  have  been  avoided, 
which  at  the  very  outset  threw  the  whole  matter  into  confusion.  The 
words  of  his  edict  establishing  the  Julian  system  have  not  been  handed 
down  to  us,  but  it  is  probable  that  they  contained  some  expression  equi* 
valent  to  "every  fourth  year,"  which  the  priests  misinterpreting  after  his 
death  to  mean  (according  to  the  sacerdotal  system  of  numeration)  as 
counting  tfte  leap  year  newly  elapsed  as  No.  1  of  the  four,  intercalated 
every  third  instead  of  every  fourth  year.  This  erroneous  practice  con- 
tinued during  86  years,  in  which  therefore  12  instead  of  9  days  were 
intercalated,  and  an  error  of  three  days  produced ;  to  rectify  which,  Au- 
gustus ordered  the  suspension  of  all  intercalation  during  three  complete 
quadriennia,  —  thus  restoring,  as  may  be  presumed  his  intention  to  have 
been,  the  Julian  dates  for  the  future,  and  re-establishing  the  Julian 
system,  which  was  never  afterwards  vitiated  by  any  error,  till  the  epoch 
when  its  own  inherent  defects  gave  occasion  to  the  Gregorian  reformation. 
According  to  the  Augustan  reform,  the  years  A.  u.  o.  761,  765,  769,  &c., 
which  we  now  call  A.  D.  8,  12,  16,  &c.,  are  leap  year  And  starting 
from  this  as  a  certain  fact,  (for  the  statements  of  the  transaction  by  clas- 
sical authors  are  not  so  precise  as  to  leave  absolutely  no  doubt  as  to  the 
previous  intermediate  years,)  astronomers  and  chronologists  have  agreed 
to  reckon  backwards  in  unbroken  succession  on  this  principle,  and  thus  to 
carry  the  Julian  chronology  into  past  time,  as  if  it  had  never  suffered 
buch  interruption,  and  as  if  it  were  certain  (which  it  is  not,  though  we  con- 
ceive the  balance  of  probabilities  to  incline  that  way')  that  Caesar,  by  way 

•  With  Scaliger,  Ideler,  and  all  the  beat  authoritiea.  Yet  it  haa  been  argued  that 
Caesar  would  naturally  begin  his  first  quadriennium  with  three  ordinary  years,  defer- 


LUNAR   OTCLE. 


531 


of  securing  the  intercaktion  as  a  matter  of  precedent,  made  bis  initiul 
year  45  B.  o.  a  leap  year.  Whenever,  therefore,  in  the  relation  of  any 
event,  either  in  ancient  history  or  in  modern,  previous  to  the  change  of 
siyle,  the  time  is  specified  in  our  modern  nomenclature,  it  is  always  to  be 
unu^rstood  as  having  been  identified  with  the  assigned  date  by  threading 
the  mazes  (often  very  tangled  and  obscure  ones)  of  special  and  national 
chronology,  and  referring  the  day  of  its  occurrence  to  its  place  in  the 
Julian  system  so  interpreted. 

(920.)  Different  nations  in  different  nges  of  the  world  have  of  course 
reckoned  their  time  in  different  ways,  and  from  different  epochs,  and  it  is 
therefore  a  matter  of  great  convenience  that  astronomers  aud  chronologists 
(as  they  have  agreed  on  the  uniform  adoption  of  the  Julian  system  of 
years  and  months)  should  also  agree  on  an  epoch  antecedent  to  them  all, 
to  which,  as  to  a  fixed  point  in  time,  the  whole  list  of  chronological  eras 
can  bn  differentially  referred.  Such  an  epoch  is  the  noon  of  the  Ist  of 
January,  B.  0.  4713,  which  is  called  the  epoch  of  the  Julian  period,  a  cycle 
of  7980  Julian  years,  to  understand  the  origin  of  which  we  must  explain 
that  of  three  subordinate  cycles,  from  whose  combination  it  takes  its  rise, 
by  the  multiplication  together  of  the  numbers  of  years  severally  contained 
in  them,  viz. : — the  Solar  and  Lunar  cycley,  and  that  of  the  indictions. 

(921.)  The  Solar  cycle  consists  of  28  Julian  years,  after  the  lapse  of 
which  the  same  days  of  the  week  on  the  Julian  system  would  alway* 
return  to  the  same  days  of  each  month  throughout  the  year.  For  four 
such  years  consisting  of  1461  days,  which  is  not  a  multiple  of  7,  it  is 
evident  that  the  least  number  of  years  which  will  fulfil  this  condition 
must  be  seven  times  that  interval,  or  28  years.  The  place  in  this  cycle 
for  any  year  A.  D.,  as  1849,  is  found  by  adding  9  to  the  year,  and  divi- 
ding by  28.     The  remainder  is  the  number  sought,  0  being  counted  as  28. 

(922.)  The  Lunar  cycle  consists  of  19  years  or  235  lunations,  which 
differ  from  19  Juliaii  years  of  SGoJ  days  only  by  about  an  hour  and  a 
half,  so  that,  supposing  the  new  moon  to  happen  on  the  first  of  January, 
in  the  first  year  of  the  cycle,  it  will  happen  on  that  day  (or  within  a  very 
short  time  of  its  beginning  or  ending)  again  after  p.  lapse  of  19  years,  and 
almost  certainly  on  that  day,  and  within  an  hour  and  a  half  of  th.  same 
hour  of  the  day,  after  the  lapse  of  four  such  cycles,  or  76  years ;  and  all 
the  new  moons  in  the  interval  will  run  on  the  same  days  of  the  month  aa 
in  the  preceding  cycle.  This  period  of  19  years  is  sometimes  called  the 
Metonic  cycle,  from  its  discoverer  Meton,  an  Athenian  mathematician,  a 

ring  the  rectification  of  their  accumulated  error  to  the  fourth,  by  inserting  there  the 
intercalary  day.  For  the  correction  of  Roman  dates  during  the  fifty-two  years  between 
the  Julian  and  Augustan  reformations,  see  Ideler,  "  Handbuch  der  Mathematischen 
und  Technischen  Chronologic,"  which  we  take  for  our  guide  throughout  this  chapter 


!l! 


m 


.•■ 


682 


OUTLINES  OF  ASTRONOMY. 


discovery  duly  appreciated  by  his  countrymen,  as  ensuring  the  oorroftpon- 
dence  between  the  lunar  and  solar  years,  the  former  of  which  was  followed 
by  the  Qroeks.  Public  honours  were  decreed  to  him  for  this  discovery, 
a  circumstance  very  expressive  of  the  annoyance  which  a  lunar  year  of 
necessity  inflicts  on  a  civilized  people,  to  whom  a  regular  and  simple 
calendar  is  one  of  the  first  necessities  of  life.  The  cycle  of  76  years,  a 
great  improvement  on  the  Metonio  cycle,  was  first  proposed  by  Callippus, 
and  is  therefore  called  the  Callippic  cycle.  To  find  the  place  of  a  given 
year  in  the  lunar  cycle,  (or  as  it  is  called  the  Golden  Number,)  add  1  to 
the  number  of  the  year  A.  D.,  and  divide  by  19,  the  remainder  (or  19  if 
exactly  divisible)  is  the  Golden  Number. 

(928.)  The  cycle  of  the  indictions  is  a  period  of  15  years  used  in  the 
courts  of  law  and  in  the  fiucul  organization  of  the  Roman  empire,  under 
Constantine  and  his  successors,  and  thence  introduced  into  legal  dates,  as 
the  Golden  Number,  serving  to  determine  Easter,  was  in  to  ecclesiastical 
ones.  To  find  the  place  of  a  year  in  the  indiction  cycle,  add  8  and  divide 
by  15.  The  remainder  (or  15  if  0  remain)  is  the  number  of  the  indic- 
tional  year. 

(924.)  If  we  multiply  together  the  numbers  28,  19,  and  15,  we  get 
7980,  ond  therefore,  a  period  or  cycle  of  7980  years  will  bring  round  the 
years  of  the  three  cycles  again  in  the  same  order,  so  that  each  year  shall 
hold  the  same  place  in  all  the  three  cycles  as  the  corresponding  year  in 
the  foregoing  period.  As  none  of  the  three  numbers  in  question  have  any 
common  factor,  it  is  evident  that  no  two  years  in  the  same  compound 
period  can  agree  in  all  the  three  particulars :  so  that  to  specify  the  numbers 
of  a  year  in  each  of  these  cycles  is,  in  fact,  to  specify  the  year,  if  within 
that  long  period;  which  embraces  the  entire  of  authentic  chronology. 
The  period  thus  arising  of  7980  Julian  years,  is  called  the  Julian  period, 
and  it  has  been  found  so  useful,  that  the  most  competent  authorities  have 
not  hesitated  to  declare  that,  througL  its  employment,  light  and  order 
were  first  introduced  into  chronology.'  We  owe  its  invention  or  revival 
to  Joseph  Scaliger,  who  is  said  to  have  received  it  from  the  Greeks  of  Con- 
stautinople  The  first  year  of  the  current  Julian  period,  or  that  of  which 
the  number  in  each  of  the  three  subordinate  cycles  is  1,  was  the  year  4718 
B.  c,  and  the  noon  of  the  1st  of  January  of  that  year,  for  the  meridian 
of  Alexandria,  is  the  chronological  epoch,  to  which  all  historical  eras  are 
most  readily  and  intelligibly  referred,  by  computing  the  number  of  integer 
days  intervening  between  that  epoch  and  the  noon  (for  Alexandria)  of  the 
day,  which  is  reckoned  to  be  the  first  of  the  particular  era  in  question. 
The  meridian  of  Alexandria  is  chosen  as  that  to  which  Ptolemy  refers  the 
contmeuoement  of  the  era  of  Nabonassar,  the  basis  of  all  his  calculations. 
'  Ideler,  Handbuch,  &c.,  vol.  1,  p.  77. 


i    I 


OHRONOLOaiOAL   ERAS. 


533 


we  get 


(925.)  0*1  veu  tho  year  of  the  Julian  period,  those  of  the  subordinato 
cycles  nro  easily  deter  mined  us  above.  Conversely,  given  tho  years  of  tho 
solar  and  lunar  cyeles,  and  of  tho  iudictiou,  to  detcriuino  the  year  of  tho 
Julian  period  proceed  as  follows :  —  Multiply  tho  number  of  tho  year  in 
the  solar  cycle  by  4845,  in  the  lunar  by  4200,  and  in  tho  Cycle  of  tho 
Indictions  by  ()91G,  divide  tho  sura  of  tho  products  by  7980,  and  tho 
remainder  is  the  year  of  the  Julian  period  sought. 

(920.)  The  following  table  contains  these  intervals  for  sorao  of  the  more 
important  historical  eras  :  — 

Intervals  in  Daij»  heticccn  the  Commencement  of  the  Julian  Period,  and 
that  of  some  other  remarkable  chronolojieal  and  astronomical  Eras. 


Numea  by  whioh  th«  Era  is  usually  cited. 


Julian  Epochs, 

Julian  period  

Crention  of  the  world  (Usher)  

Era  of  tlio  Deluge  (AboulhiisBiin  Kua- 

ohiar) 

Ditto  Vulvar  Computation 

Era  of  Abraham  (Sir  II.  Nicholas) .... 

Destruction  of  Troy,  (ditto) 

Dudiuatiim  of  Solomon's  Temple 

Olympiads  (mean  epoch  in  general 

use)  

Building;  of  Home  (Varroniau  epoch, 

u.  c.) 

Era  of  NabonasKur 

Motoiiie  cycle  (AMtronomiciil  epoch)... 

Callippic  cycle,         do.     (Hiot) 

Pliilippic  era,  or  era  of  Philip  Aridieus 

Era  of  tho  Seleucidro 

Cwsareau  era  of  Antioch 

Julian  reformation  of  tho  Calendar... 

Spanish  era 

Actian  era  in  Rome 

Actian  era  of  Alexandria 

Vulgar  or  Dionysian  era 

Era  of  Diocletian 

Uejira    (astronomical    epoch,    new 

moon) 

Era  of  Yezdogird 

Qelalrean  era  (Sir  II.  Nicholas) 

Last  day  of  Old  Style  (CathcUc  no- 

tions) 

Last  day  of  Old  Style  in  England).... 

Gregorian  Epoclm. 
New  Style  in  Catholic  nations  

Ditto      in  England 

Commencement  of  the  19th  century, 

epoch  of  Bode's  catalogue  of  stars., 
Epoch  of  the  catalogue  of  stars  of  the 

R.  Astronomical  Society 

Epoch  of  the  catalogue  of  the  British 

Association  


Julian  JJatet. 


First  clny 

current  of 

tho  era. 


Chronoloi?*! 
UuRiKDatloD 
of  the  year. 


Ctf rent  year 
of  the  J  u- 
■an  i'erioU. 


Jan.  1. 

B.C.  4713 

1   1 

(Jan.  1.) 

4UU4 

710 

Feb.  18. 

3102 

U\i 

(Jan.  1.) 

234'. 

2366 

Oct.  1. 

201  > 

2699 

July  12. 

lU. . 

8530 

(May  1.) 

1015 

3699 

July  1. 

776 

3938 

April  22. 

753 

3961 

Feb.  2(1. 

747 

3967 

July  15. 

43J 

4282 

June  28. 

330 

4384 

Nov.  12. 

324 

4390 

Oct.  1. 

312 

4402 

Sept.  1. 

49 

4666 

Jan.  1. 

45 

4669 

Jan.  1. 

38 

4676 

Jan.  1. 

30 

4684 

Au,.  ?9. 

30 

4684 

t'i  ,1.  \. 

A.D.   1 

4714 

Aug.  2tf. 

284 

4997 

July  15. 

62" 

6335 

June  16. 

632 

5345 

March  14. 

1079 

5792 

Oct.  4. 

1582 

6296 

Sept.  2. 

1752 

6466 

Gregorian 

Dates. 

Oct.  15. 

1582 

6295 

Sept.  14. 

1752 

6466 

Jan.  1. 

1801 

6514 

Jan.  1. 

1830 

6543 

Jan.  1. 

1850 

656.1 

Interval 
days. 


0 

268,963 

688/ -e 

863,   ;• 

98J,V18 

1,289,160 

1,350,815 

1,438,171 

l,44ft,,502 
1,448,638 
l,5t)3,.S31 
1,69!>,608 
l,60:{,;i!)8 
1,007,7:'.0 
1,703,770 
1,704,987 
1,707,544 
1,710,466 
1,710,706 
1,721,424 
1,825,030 

1,948,439 
1,952,063 
2,115,285 

2,299,160 
2,361,221 


2,209,161 
2,361,222 

2,378,862 

2,389,454 

2,396,769 


'^il: 


( 


534 


OUTLINES   OF  ASTRONOMY. 


N.  B.  The  oiril  epochs  of  the  Metonio  cycle,  and  the  Hejira,  are  each  one  day  later 
than  the  astronomical,  the  latter  being  the  epochs  of  the  absolute  neto  mooui,  the  former 
those  of  the  earliest  possible  visibility  of  the  lunar  crescent  in  a  tropical  sky.  M.  Biot 
has  shown  that  the  solstice  and  new  moon  not  only  coincided  on  the  day  here  set  down 
as  the  commencement  of  the  Callippio  cycle,  but  that,  by  a  happy  coincidence,  a  bare 
possibility  existed  of  seeing  the  orescent  moon  at  Athena  loitAtn  that  day,  reckoned  from 
midnight  to  midnight. 


(927.)  The  determination  of  the  exact  interval  between  any  two  given 
dates  is  a  matter  of  such  importance,  and,  unless  methodically  performed, 
is  so  very  liable  to  error,  that  the  following  rules  will  not  be  found  out  of 
place.  In  the  first  place  it  must  be  remarked,  generally,  that  a  date, 
whether  of  a  day  or  year,  always  expresses  the  day  or  year  airrent  and  not 
elapsed,  and  that  the  designation  of  a  year  by  A.  D.  or  B.  c.  is  to  be  re- 
garded as  the  name  of  that  year,  and  not  as  a  mere  number  uninterrupt- 
edly designating  the  place  of  the  year  in  the  scale  of-  time.  Thus,  in 
the  date,  Jan.  5,  3.  C.  1,  Jan.  5  docs  not  mean  that  5  days  of  January  in 
the  year  in  question  have  elapsed,  but  that  4  have  elapsed,  and  the  5th  is 
current.  And  the  B.  C.  1,  indicates  that  the  first  day  of  the  year  so 
named,  (the  first  year  current  before  Christ,)  preceded  the  first  day  of  the 
vulgar  era  by  one  year.  The  scale  of  a.  d.  and  b.  c.  is  not  continuous, 
the  year  0  in  both  being  wanting;  so  that  (supposing  the  vulgar  reckon- 
ing correct)  our  Saviour  was  born  in  the  year  b.  c.  1. 

(928.)  To  find  the  year  current  of  the  Julian  period,  (j.  P.)  corre- 
sponding to  any  given  year  current  B.  C.  or  A.  D.  If  B.  C,  subtract  the 
number  of  the  year  from  4711 :  if  A.  d.,  add  its  number  to  4713.  For 
example,  see  the  foregoing  table. 

(929.)  To  find  the  day  current  of  the  Julian  period  corresponding  to 
any  given  date,  Old  Style.  Convert  the  year  B.  c.  or  A.  D.  into  the  cor- 
responding year  J.  p.  as  above.  Subtract  1  and  divide  the  number  so 
diminished  by  4,  and  call  Q  the  integer  quotient,  and  B  the  remainder. 
Then  will  Q  be  the  number  of  entire  quadriennia  of  1461  days  each,  and 
R  the  residual  years,  th^  first  of  which  is  always  a  leap-year.  Convert 
Q  into  days  by  the  help  of  the  first  of  the  annexed  tables,  and  R  by  the 
second,  and  the  sum  will  be  the  interval  between  the  Julian  epoch,  and 
the  commencement,  Jan.  1,  of  the  year.  Then  find  the  days  intervening 
between  the  beginning  of  Jan.  1,  and  that  of  the  date-day  by  tlie  third 
table,  using  the  column  for  a  leap-year,  where  R  =  0,  and  that  for  a  com- 
mon year  when  R  is  1,  2,  or  3.  Add  the  days  so  found  to  those  in 
Q  -f  R,  and  the  sum  will  be  the  days  elapsed  of  the  Julian  period,  the 
number  of  which  increased  by  1  gives  the  day  current. 


'  / 


:.JiJ 


CHRONOLOGICAL  INTERVALS. 


635 


Table  I.  Multiples  of  1461, 

the  days  in  a 

Table  2.  Days  in 
Residual  years. 

juiian  ijuaavtenmum. 

0 

1 
2 
3 

0        1 

1 
2 
3 

1461 
2922 
4383 

4 
5 
6 

5844 
7305 
8766 

7 
8 
9 

10227 
11688 
13149 

366 

731 

1096 

Table  3.  Days  elapsed  from  Jan.  1  to  the  1st  of  each  Month. 

In  a  com- 
mon year. 

In  a  leap 
year. 

In  a  com- 
mon year. 

In  a  leap 
year. 

Jan.  1 

0 

31 

59 

90 

120 

161 

0 

31 

60 

91 

121 

162 

July  1 

181 
212 
243 
273 
304 
334 

182 
213 
244 
274 
305 
335 

Feb.  1 

Aug.  1 

March  1 

Sept.  1 

April  1 

Oct.  1 

May  1 

Nor.  1 

June  1 

Dec.  1 

Example. — What  is  the  current  day  of  the  Julian  period  correspond- 
ing to  the  last  day  of  Old  Style  in  England,  on  Sept.  2,  A.  D.  1752  ? 


1752 
4713 

6465  year  current. 
1 


4)6464  yeara  elapsed. 
Q=16167 
R=>     o5 


1000 

1,461,000 

600 

876,600 

10 

14,610 

6 

8,766 

R  =  0 

0 

Jan.  1  to  Sept. 

1, 

244 

Sept.  1  to  Sept. 

2, 

1 

2,361,221  days  elapsed. 


Current  day  the  2,361,222". 

(930.)  To  find  the  same  for  any  given  date,  New  Style.  Proceed  as 
above,  considering  the  date  as  a  Julian  date,  and  disregarding  the  change 
of  style.     Then,  from  the  resulting  days,  subtract  as  follows : — 

For  any  date  of  New  Style,  antecedent  to  March  1,  a.  d.  1700 10  days. 

After  Feb.  28,  1700,  and  before  March  1,  a.  d.  1800 11  days. 

"  1800  "  "  1900 12  days. 

••  1900  '*  "  2100 13  days,  &c. 

(931.)  To  find  the  interval  between  any  two  dates,  whether  of  Old  or 
New  Style,  or  one  of  one,  and  one  of  the  other.  Find  the  day  current 
of  the  Julian  period  corresponding  to  each  date,  and  their  difference  is 
the  interval  required.  If  the  dates  contain  hours,  minutes,  and  seconds, 
they  must  be  annexed  to  their  respective  days  current,  and  the  subtraction 
performed  as  usual. 

(932.)  The  Julian  rule  made  every  fourth  year,  without  exception,  a 
bissextile.  This  is,  in  fact,  an  over-correction ;  it  supposes  the  length  of 
the  tropical  year  to  be  365J*,  which  is  too  great,  and  thereby  induces  an 


Mi 


:  •    i 


\ri 


III 


iriii 


536 


OUTLINES   OF  ASTRONOMY. 


error  of  7  days  in  908  years,  as  will  easily  appear  on  trial.  Accordingly, 
so  early  as  the  year  1414,  it  began  to  be  perceived  that  the  equinoxes 
were  gradually  creeping  away  from  the  21st  of  March  and  September, 
where  they  ought  to  have  always  fallen  had  the  Julian  year  been  exact, 
and  happening  (as  it  appeared)  too  early.  The  necessity  of  a  fresh  and 
effectual  reform  in  the  calendar  was  from  that  time  continually  urged,  and 
at  length  admitted.  The  change  (which  took  place  under  the  popedom 
of  Gregory  XIII.)  consisted  in  the  omission  of  ten'  nominal  days  after 
the  4th  of  October,  1582,  (so  that  the  next  day  was  called  the  15th,  and 
not  the  5th,)  and  the  promulgation  of  the  rule  already  explamed  for  future 
regulation.  The  change  was  adopted  immediately  in  all  Catholic  coun- 
tries; but  more  slowly  in  Protestant.  In  England,  "the  change  of  style," 
as  it  was  called,  took  place  after  the  2d  of  September,  1752,  eleven  no- 
minal days  being  then  struck  out ;  so  that,  the  last  day  of  Old  Style  being 
the  2d,  the  first  of  New  Style  (the  next  day)  was  called  the  14th,  instead 
of  the  3d.  The  same  legislative  enactment  which  established  the  Gre- 
gorian year  in  England  in  1752,  shortened  the  preceding  year,  1751,  by 
a  full  quarter.  Previous  to  that  time,  the  year  was  held  to  begin  with 
the  25th  March,  and  the  year  A.  d.  1751  did  so  accordingly ;  but  that 
year  was  not  suffered  to  run  out,  but  was  supplanted  on  the  1st  of  January 
by  the  year  1752,  which  it  was  enacted  should  commence  on  that  day,  as 
well  as  every  subsequent  year.  Russia  is  now  the  only  country  in 
Europe  in  which  the  Old  Style  is  still  adhered  to,  and  (another  secular 
year  having  elapsed)  the  difference  between  the  European  and  Russian 
dates  amounts,  at  present,  to  12  days. 

(983.)  It  is  fortunate  for  astronomy  that  the  confusion  of  dates,  and 
the  irreconcilable  contradctions  which  historical  statements  too  often  ex- 
hibit, when  confronted  with  the  best  knowledge  we  possess  of  the  ancient 
reckonings  of  time,  affect  recorded  observations  but  little.  An  astrono- 
mical observation,  of  any  striking  and  well-marked  phajnonienon,  carries 
with  it,  in  most  cases,  abundant  means  of  recovering  its  exact  date,  when 
any  tolerable  approximation  is  afforded  to  it  by  chronological  records; 
and,  .so  far  from  being  abjectly  dependent  on  the  obscure  and  often  con- 
tradictory dates,  which  the  comparison  of  ancient  authorities  indicates,  is 
often  itself  the  surest  and  most  convincing  evidence  on  which  a  chrono- 
logical epoch  can  be  brought  to  rest.  Remarkable  eclipses,  for  instance, 
now  that  the  lunar  theory  is  thoroughly  understood,  can  be  calculated 
back  for  several  thousands  of  years,  without  the  possibility  of  mistaking 
the  day  of  their  occurrence.     And,  whenever  any  such  eclipse  is  so  inter- 


WOV( 

even 

the 

date 

feet 


'  See  note  at  the  end  of  this  chapter,  p.  540. 


m 


LUNAR  YEAR. 


537 


woven  with  the  account  given  by  an  ancient  author  of  some  historical 
event,  as  to  indicate  precisely  the  interval  of  time  between  the  eclipse  and 
the  event,  and  at  the  same  time  completely  to  identify  the  eclipse,  that 
date  is  recovered  and  fixed  for  ever.' 

(934.)  The  days  thus  parcelled  out  into  years,  the  next  step  to  a  per- 
fect knowledge  of  time  is  to  secure  the  identification  of  each  day,  by  im- 
posing on  it  a  name  universally  known  and  employed.  Since,  however, 
the  days  of  a  whole  year  are  too  numerous  to  admit  of  loading  the 
memory  with  distinct  names  for  each,  all  nations  have  felt  the  necessity 
of  breaking  them  down  into  parcels  of  a  more  moderate  extent ;  giving 
names  to  each  of  these  parcels,  and  particularizing  the  days  in  each  by 
numbers,  or  by  some  special  indication.  The  lunar  month  has  been  re- 
sorted to  in  many  instances ;  and  some  nations  have,  in  fact,  preferred  a 
lunar  to  a  solar  chronology  altogether,  as  the  Turks  and  Jews  continue  to 
do  to  this  day,  making  the  year  consist  of  12  lunar  months,  or  354  days. 
Our  own  division  into  twelve  unequal  months  is  entirely  arbitrary,  and 
often  productive  of  confusion,  owing  to  the  equivoque  between  the  lunar 
and  calendar  month.'  The  intercalary  day  naturally  attaches  itself  to 
February  as  the  shortest. 

(935.)  Astronomical  time  reckons  from  the  noon  of  the  current  day, 
civil  f**:  n  the  preceding  midnight,  so  that  the  two  dates  coincide  only 
durLdg  the  earlier  half  of  the  astronomical,  and  the  later  of  the  civil  day. 
Tims  is  an  inconvenience  which  might  be  remedied  by  shifting  the  astro- 
nomical epoch  to  coincidence  with  the  civil.  There  is,  however,  another 
inconvenience,  and  a  very  serious  one,  to  which  both  are  liable,  inherent 
in  the  nature  of  the  day  itself,  which  is  a  local  phaenomenon,  and  com- 
mences at  different  instants  of  absolute  time,  under  different  meridians, 
whether  we  reckon  from  noon,  midnight,  sunrise,  or  sunset.  In  conse- 
quence all  astronomical  observations  require,  in  addition  to  their  date,  to 
render  them  comparable  with  each  other,  the  longitude  of  the  place  of 
observation  from  some  meridian,  commonly  respected  by  all  astronomers. 
For  geographical  longitudes,  the  Isle  of  Ferroe  has  been  chosen  by  some 
as  a  common  meridian,  indifferent  (and  on  that  very  account  offensive)  to 
all  nations.  Were  astronomers  to  follow  such  an  example,  they  would 
probably  fix  upon  Alexandria,  as  that  to  which  Ptolemy's  observations 


m 


'  See  the  remarkable  calculations  of  Mr.  Baily  relative  to  the  celebrfiiod  solar  eclipse 
which  put  an  end  to  the  battle  between  the  kings  of  Media  and  Lyilia,  b.  c.  610, 
Sept.  30.    Phil.  Trans,  ci.  220. 

'  "A  month  in  law  is  a  lunar  month  or  twenty-eight  days,  (! !)  unless  otherwise  ex- 
pressed." — Blackslone,  ii,  chap.  9.  "A  lease  for  twelve  months  is  only  for  forty-eight 
weeks."— iiid. 


538 


OUTLINES   OF  ASTRONOMY. 


and  computations  were  reduced,  and  as  claiming  on  that  account  the  re- 
spect of  all  while  offending  the  national  egotism  of  none.  But  even  this 
will  not  meet  the  whole  diflSculty.  It  will  still  remain  doubtful,  on  a 
meridian  180°  remote  from  that  of  Alexandria,  what  day  is  intended  by 
any  given  date.  Do  what  we  will,  when  it  is  the  Monday,  the  1st  of 
January,  1  ^  ^9,  in  one  part  of  the  world,  it  will  be  Sunday,  the  31st  of 
December,  1848,  in  another,  so  long  as  time  is  reckoned  by  local  hours. 
This  equivoque,  and  the  necessity  of  specifying  the  geographical  locality 
as  an  eleuout  of  the  date,  can  only  be  got  over  by  a  reckoning  of  time 
which  refers  itself  to  some  event,  real  or  imaginary,  common  to  all  the 
globe.  Such  an  event  is  the  passage  of  the  sun  through  the  vernal 
equinox,  or  rather  the  passage  of  an  imaginary  sun,  supposed  to  move 
with  perfect  equality,  through  a  vernal  equinox  supposed  free  from  the 
inequalities  of  nutation,  and  receding  upon  the  ecliptic  with  perfect  uni- 
formity. The  actual  equinox  is  variable,  not  only  by  the  effect  of  nuta- 
tion, but  by  that  of  the  inequality  of  precession,  resulting  from  the  change 
in  the  plane  of  the  ecliptic  due  to  planetary  perturbation.  Both  vai-ia- 
tions  are,  however,  periodical ;  the  one  in  the  short  period  of  19  years, 
the  other  in  a  period  of  enormous  length,  hitherto  uncalculated,  and 
whose  maximum  of  fluctuation  is  also  unknown.  This  would  appear,  at 
first  sight,  to  render  impracticable  the  attempt  to  obtain  from  the  sun's 
motion  any  rigorously  uniform  measure  of  time.  A  little  consideration, 
however,  will  satisfy  us  that  such  is  not  the  case.  The  solar  tables,  by 
which  the  apparent  place  of  the  sun  in  the  heavens  is  represented  with 
almost  absolute  precision  from  the  earliest  ages  to  the  present  time,  are 
constructed  upon  the  supposition  that  a  certain  angle,  which  is  called  "  the 
sun's  mean  longitude,"  (and  which  is,  in  effect,  the  sum  of  the  mean  side- 
real motion  of  the  sun,  plxis  the  mean  sidereal  motion  of  the  equinox  in 
the  opposite  direction,  as  near  as  it  can  be  obtained  from  the  accumu- 
lated observations  of  twenty-five  centuries,)  increases  with  rigorous  uni- 
formity as  time  advances.  The  conversion  of  this  mean  longitude  into 
ti»ne  at  the  rate  of  360°  to  the  mean  tropical  year,  (such  as  the  tables 
assume  it,)  will  therefore  give  us  both  the  unit  of  time,  and  the  uniform 
measure  of  its  lapse  which  we  seek.  It  will  also  furnish  us  with  an  epoch, 
not  indeed  marked  by  any  real  event,  but  not  on  that  account  the  less 
positively  fixed,  being  connected,  through  the  medium  of  the  tables,  with 
every  single  observation  of  the  sun  on  which  they  have  been  constructed 
and  with  which  compared. 

(936.)  Such  is  the  simplest  abstract  conception  of  equinoctial  time.  It 
ia  the  mean  longitude  of  the  sun  of  some  one  approved  set  of  solar  tables, 
converted  into  time  at  the  rate  of  300°  to  the  tropical  year.     Its  unit  is 


ue^'S  J^    ,*i 


EQUIXOCTIAL  TIME. 


539 


the  mean  tropical  year  which  those  tables  assume  and  no  other,  and  its 
epoch  is  the  mean  vernal  equinox  of  these  tables  for  the  current  year,  or 
the  instant  when  the  mean  longitude  of  the  tables  is  rigorously  0,  accord- 
iug  to  the  assumed  mean  motion  of  the  sun  and  equinox,  the  assumed 
epoch  of  mean  longitude,  and  the  assumed  equinoctial  point  on  which  the 
tables  have  been  computed,  and  no  other.  To  give  complete  effect  to  this 
idea,  it  only  remains  to  specify  the  particular  tables  fixed  upon  for  the 
purpose,  which  ought  to  be  of  great  and  admitted  excellence,  since,  once 
decided  on,  ^be  very  essence  of  the  conception  is  that  no  subsequent  alterca- 
tion in  any  respect  should  he  made,  even  when  the  continual  progress  of 
astronomical  science  shall  have  shown  any  one  or  all  of  the  elements  con- 
cemed  to  be  in  smne  minute  degree  erroneous  (as  necessarily  they  must,) 
and  shall  have  even  ascertained  the  corrections  they  require  (to  be  them- 
selves again  corrected,  when  another  step  in  refinement  shall  have  been 
made.) 

(937.)  Delambre's  solar  tables  (in  1828)  when  this  mode  of  reckoning 
time  was  first  introduced,  appeared  entitled  to  this  distinction.  According 
to  these  tables,  the  sun's  mean  longitude  was  C,  or  the  mean  vernal  equi- 
nox occurred,  in  the  year  1828,  on  the  22d  of  March  at  1'  2"  59*  05 
mean  time  at  Greenwich,  and  therefore  at  1"  12™  20*-55  mean  time  at 
Paris,  or  1"  SG"  34"*55  mein  time  at  Berlin,  at  which  instant,  therefore, 
the  equinoctial  time  was  0*  0"  0"  ©••OO,  being  the  commencement  ot'  the 
1828th  year  current  of  equinoctial  time,  if  we  choose  to  date  from  the 
mean  tabular  equinox,  nearest  to  the  vulgar  era,  or  of  the  6541  st  year  of 
the  Julian  period,  if  we  prefer  that  of  the  first  year  of  that  period. 

(938.)  Equinoctial  time  then  dates  from  the  mean  vernal  equinox  of 
Delambre's  solar  tables,  and  its  unit  is  the  mean  t/opical  year  of  these 
tables  (365**242264.)  Hence,  having  the  fractionil  part  of  a  day  ex- 
pressing the  difference  between  the  mean  local  time  at  any  place  (suppose 
Greenwich)  on  any  one  day  between  two  consecutivi^  mean  vernal  equi- 
noxes, that  difference  will  be  the  same  for  every  other  day  in  the  same 
interval.  Thus,  between  the  mean  equinoxes  of  1828  and  1829,  the 
difference  between  equinoctial  and  Greenwich  time  is  0*956261  or  O"" 
22*  57"  0»-95,  which  expresses  the  equinoctial  day,  hour,  minute,  and 
second,  corresponding  to  mean  noon  at  Greenwich  on  March  23,  1828, 
and  for  the  noons  of  the  24th,  25th,  &c.,  we  have  only  to  substitute  Id, 
2d,  &c.,  for  0*,  retaining  the  same  decimals  of  a  day,  or  the  same  hours, 
minutes,  &c.,  up  to  and  including  March  22, 1829.  Between  Greenwich 
noon  of  the  22d  and  23d  of  March,  1829,  the  1828th  equinoctial  year 
terminates,  and  the  1829th  commences.  This  happens  at  0''-286003,  or 
at  Qi^  51™  50»-66  Greenwich  mean  time,  after  which  hour,  and  until  the 


ill 


540 


OUTLINES  OP  ASTRONOMY. 


next  noon,  the  Greenwich  hour  added  to  equinoctial  time  364*-956261 
will  amount  to  more  than  865*242264,  a  complete  year,  which  has  there- 
fore to  be  subtracted  to  get  the  equinoctial  date  in  the  next  year,  cor- 
responding to  the  Greenwich  time.  For  exarjsr.ie,  at  12''  0"  0*  Greenwich 
U3«an  time,  or  O'-SOOOOO,  the  equinoctial  timo  will  !'«.  36vi56261  + 
0-500000=i;05-456261,  which  being  gnawer  hm  86iv2422(i;S  shows 
that  the  equinoctial  year  current  has  changed,  and  the  laik>  number 
being  oubtracted,  we  get  C'''-213U77  for  iJio  oqumoocial  tiriu  of  the 
1829th  year  cuirent  coneepoading  to  March  22,  12"  Greenwich  moan 
time. 

(939.)  Having,  therefore,  she  fractional  part  of  a  day  for  a'.:iy  -^ne  year 
expressing  the  equinoctial  hour,  Ac,  at  thp  mean  noon  of  ;<ri/  given  place, 
that  for  succeeding  j  iars  will  bo  had  by  subtracting  0*'242264,  and  its 
multiples,  from  such  fractional  part  (increased  if  nece  u  y  by  unity,)  and 
for  preceding  years  by  adding  them.  Thns,  havT.ag  found  0-198525  for 
the  ixactional  ptat  for  1827,  we  find  for  the  fractional  parts  for  succeeding 
years  up  to  185C  as  follows': — 

•110981 
•868717 
•626453 
•384189 
•141925 
•899661 


"  These  numbers  differ  from  those  in  the  Nautical  Almanack,  and  would  require  to 
be  substituted  for  them,  to  carry  out  the  idea  of  equinoctial  time  as  above  laid  down. 
In  the  years  1828-1833,  the  late  eminent  editor  of  that  work  used  an  equinox  slightly 
differing  from  that  of  Delambre,  which  accounts  for  the  difference  in  those  years.  In 
1834,  it  would  appear  that  a  deviation  both  from  the  principle  of  the  text  and  from  the 
previous  practice  of  that  ephemeris  took  place,  in  deriving  the  fraction  for  1834  from 
that  for  1833,  which  has  been  ever  since  perpetuated.  It  consisted  in  rejecting  the 
mean  longitude  of  Delambre's  tables,  and  adopting  Bessel's  correction  of  that  element. 
The  effect  of  this  alteration  was  to  insert  3"  3"68  of  purely  imaginary  lime,  between 
the  end  of  the  equinoctial  year  1833  and  the  beginning  of  1834,  or,  in  other  words,  to 
make  the  interval  between  the  noons  of  March  22  and  23,  1834,  24''  3«>  3i'68,  when 
reckoned  by  equinoctial  time.  In  1835,  and  in  all  subsequent  years,  a  further  depar- 
ture from  the  principle  of  the  text  took  place  by  substituting  Bessel's  tropical  year  of 
365*2422175,  for  Delambre's.    Thus  the  whole  subject  has  fallen  into  confusion. 

[Note  on  Art.  932. 
The  reformation  of  Gregory  was,  after  all,  incomplete.  Instead  of  10  days  he  ought 
to  have  omitted  12.  The  interval  from  Jan.  1,  a.  d.  1,  to  Jan.  1,  a.d.  1582,  reckoned 
as  Julian  years,  is  577460  days,  and  ns  tropical,  577448,  with  an  error  not  exceeding 
0^*01,  the  difference  being  12  days,  whose  omission  would  have  completely  restored  the 
•  ulian  epoch.  But  Gregory  assumed  for  his  fixed  point  of  departure,  not  that  epoch, 
but  one  later  by  324  years,  viz.  Jan.  1,  a.d.  325,  the  year  of  the  Council  of  Nice; 
assuming  which,  the  difference  of  the  two  reckonings  is  9''505,  or,  to  the  nearest  whole 
number,  10  days.] 


j>m 

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1842 

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1843 

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1849 

1830 

•471733 

1837 

•775885 

1844 

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1860 

1831 

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1838 

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1645 

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1832 

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1862 

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APPENDIX. 


548 


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544 


APPENDIX. 


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APPENDIX. 


15 


Note. — The  elements  of  the  orbits  of  M«rcury,  Venun,  the  Earth,  Mar«,  Jupitor, 
Saturn,  and  Urnnus,  are  those  given  by  the  late  F.  Daily,  Esq.,  in  his  "  Astronomical 
Tables  and  Formulae,"  and  are  the  same  with  those  which  form  the  basis  of  Dclam> 
bro's  tables,  embodying  the  forniulo)  of  Laplace.  The  elements  of  Uranus  and  Nep< 
tune  can  only  bo  regarded  as  provisional;  those  of  the  former  requiring  considerable 
corrections,  necessitated  by  the  discovery  of  Neptune,  but  which,  not  being  yet  finally 
ascertained,  by  reason  of  the  uncertainty  still  attending  on  the  mass  and  elements  of 
the  latter  planet,  it  was  thought  better  to  linve  the  old  elements  untouched  than  to  give 
nn  imperfect  rectification  of  them.  I'he  masses  of  the  planets  are  those  most  recently 
adopted  by  Encke  (Ast.  Nachr.  No.  443),  on  mature  consideration  of  all  the  autho- 
rities, that  of  Neptune  excepted,  which  is  Prof.  Pierce's  determination  from  Bund's 
and  Lajisell's  observation  of  the  satellite  discovered  by  the  latter.  Th«  densities  are 
Hansen's  (A.N.  443.) 

The  elements  of  Vesta,  Juno,  Ceres,  and  Pallas,  are  the  osculating  elements  for 
1850,  computed  by  Encke  (A.  N.  636.)  [Those  of  Flora  are  from  the  compuJoiions 
of  Brunnow  (A.  N.  645);  of  Victoria,  Villar.  aux  (A.  N.  711);  of  Iris,  Schube/s 
(A.  N.  730) ;  of  Metis,  Wolfers  ^A.  N.  764) ;  of  Hebe,  Luther  (A.  N.  721) ;  of  Par- 
thenope,  Galen  (A.  N.  757);  of  Astrsea,  D' Arrest  (A.  N.  626);  of  Egeria,  D' Arrest 
(A.  N.  749);  of  Irene,  Vogel  and  Riimker  (A.  N.  765);  and  of  Hygeia,  Santini  (A. 
N.  702.) 

Of  these  last-mentioned  small  planets,  Hygeia,  Porthenope,  and  Egeria  were  dis- 
covered by  Dr.  Gasparis,  at  Naples,  on  April  12,  1849,  May  11  and  Nov.  2,  1850, 
respectively;  Iris,  Flora,  Victoria,  and  Irene,  by  Mr.  Hind,  on  Aug.  13  and  Oct.  18, 
1847,  Sept.  13,  1850,  and  May  19,  1851,  respectively.  The  elements  of  the  recently- 
discovered  small  planets  may  undergo  material  corrections  from  further  observation. 
Irene  has  a  blue  colour  and  a  faint  nebulous  envelope.  The  orbits  of  Astrtea  and 
Hygeia  approach  at  one  point  (their  common  node)  wiihin  0*006  of  the  radius  of  the 
earth's  orbit.  It  will  not  be  long  before  the  planets  themselves  come  within  that  prox- 
imiiy  to  each  other  (A.  N.  752.)  Victoria  and  Astroea  are  subject  to  variations  of 
brightness,  which  indicate  rotations  on  their  a.xes,  and  dark  spots  (A.  N.  760.)  D' Arre-st 
{k.  N.  752)  remarks  that  a  relation  subsists  between  the  excentricities  of  the  orbits  of 
the  small  planets,  and  the  inclinations  of  the  planes  in  which  they  lie  to  the  sun's 
equator,  the  more  excentric  orbits  being  the  more  inclined.  While  these  sheets  pass 
through  the  press,  another,  yet  unnamed,  is  announced  by  M.  de  Gasparis.] 

III. 

Sy.NOPTio  Table  op  the  Elements  op  the  Orbits  op  the  Satel- 
lites, so  FAR  AS  they  ARE  KNOWN. 

1.  The  Moon. 

Mean  distance  from  earth .  -   59''964350OO 

Mean  sidereal  revolution 27''-321661418 

Mean  synodical ditto 29^530588715 

Excentricity  of  orbit 0054844200 

Mean  revolution  of  nodes 6793'»-391060 

Mean  revolution  of  apogee 3232''-575343 

Mean  longitude  of  node  at  epoch 13°  53'  17""7 

Mean  longitude  of  perigee  at  do 266    10     7  '5 

Mean  inclination  of  orbit 5     8  47  "9 

Mean  longitude  of  moon  at  epoch 118    17     8*3 

Mass,  that  of  earth  being  1, 0011399 

Diameter  in  miles 2153 

Density,  that  of  the  earth  bsing  1 0*5657 

'  The  distances  are  expressed  in  equatorial  radii  of  the  primaries.   The  epoch  is  Jan.  1 , 
1801,  unless  otherwise  expressed.    The  periods,  &c.  are  expressed  in  mean  solar  days. 
35 


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INDEX. 


N.  B.  Th«  referenoos  are  to  the  articles,  not  to  the  pages.  * 

...  attached  to  a  reference  number  indicates  that  the  reference  extends  to  the  article  cited,  and 
several  subsequent  in  succession. 


A. 


Aberration  of  light  explained,  329.  Its 
uranographical  effects,  338.  Of  an 
object  in  motion,  335.  How  distin- 
guished from  parallax,  805.  System- 
atic, 862. 

Ahoul  Wcfa,  705. 

Acceleration,  secular,  of  moon's  mean 
motion,  740. 

Adams,  506.  767. 

Adjustment,  errors  of,  in  instruments, 
136.  Of  particular  instruments.  (<Sce 
those  instruments.) 

^tna,  portion  of  earth  visible  from,  32. 
Height  of,  32.  note. 

Air,  rarefaction  of,  33.  Law  of  density, 
37.  Refractive  power  affected  by 
moisture,  41. 

Airy,  G.  B.  Esq.,  his  results  respecting 
figure  of  the  earth,  220.  Researches 
on  perturbations  of  the  earth  by  Ve- 
nus, 726.  Rectification  of  the  mass 
of  Jupiter,  757. 

Algol,  821. 

AUilude  and  azimuth  instrument,  187. 
— 3.  Equal,  method  of,  188. 

Andromeda,  nebula  in,  874. 

Angk  of  position,  204.  Of  situation,  311. 

Angles,  measurement  of,  163.  167.  Hour, 
107. 

Angular  velocity',  law  of,  variation  of, 
350. 

Anomalistic  year,  384. 

Anomaly  of  a  planet,  499. 

Annular  nebula?,  875. 

Apex  of  aberration,  b43.  Of  parallax, 
343.  Of  refraction,  343.  Solar,  854. 
Of  shooting  stars,  902.  904. 

Aphelion,  308. 

Apogee  of  moon,  406.  Period  of  its  re- 
volution, 687. 

Apsides,  406.  Motion  of  investigated, 
67'>.  Application  to  lunar,  076 ... 
Motion  of,  illustrated  by  experiment, 


692.  Of  planetary  orbits,  694.  Li- 
bration  of,  694.  Motion  in  orbits 
very  near  to  circles,  696.  In  excen- 
trio  orbits,  697... 

Areas,  Kepler's  law  of,  490. 

Argelander,  his  researches  on  variable 
stars,  820...,  on  sun's  proper  motion, 
854. 

Argo,  nebulas  in,  887.  Irregular  star  n 
in  constellation,  830. 

Ascension,  right,  108.  (See  Right  ascen- 
sion. ^ 

Asteroids,  their  existence  suspected  pre- 
vious to  their  discovery,  505.  Ap- 
pearance in  telescopes,  525.  Gravity 
on  surface  of,  625.  Elements,  Appen- 
dix, Synoptic  Table. 

Astrcea,  discovery  of,  505. 

Astromeier,  783,  784. 

Astronomy.  Etymology,  11.  General 
notions,  11. 

Atmosphere,  constitution  of,  33...  Possi- 
ble limit  of,  36.  Its  waves,  37.  Strata, 
37.  Causes  refraction,  38.  Twi- 
light, 44.  Total  mass  of,  148.  Of 
Jupiter,  513. 

Attraction  of  a  sphere,  445 — 450.  (See 
Gravitation.) 

Augmentation  of  moon's  apparent  dia 
meter,  404. 

Augustus,  his  reformation  of  mistakes 
in  the  Julian  calendar,  (919).  Era 
of,  926. 

Australia,  excessive  summer  tempera- 
ture  of,  369. 

Axis  of  the  earth,  82.  Rotation  porma' 
nent,  56.  Miyor  of  the  earth'^  orbit, 
373.     Of  sun's  rotation,  392. 

Axis  of  a  planetary  orbit.  Momentary 
variation  of,  caused  by  the  tangential 
force  only,  058.  600.  Its  variations 
periodical,  001...  Invariability  of, 
and  how  understood,  008. 

Azimuth,  103. — and  altitude  instrument, 

187. 

(549) 


550 


INDEX. 


B. 

Baronuiter,  nature  of  its  indication,  33. 

Uae  in  calculating  refraction,  43.    In 

determining  heights,  287. 
Belts  of  Jupiter,  612,     Of  Saturn,  514. 
Benzenberg^a   principle    of   collimation, 

179. 
Bessfil,  his  results  respecting  the  figure 

of  the  earth,  220.     Discovers  parallax 

of  61  Cygni,  812. 
Biela's  comet,  679... 
Biot,  his  aeronautic  ascent,  32. 
Bode,  his  (so  called)  law  of  planetary 

distances,  605.     Violated  in  the  case 

of  Neptune,  607. 
Borda,  his  principle  of  repetition,  198. 
Bouvard,   his   suspicion   of  extraneous 

influence  on  Uranus,  760. 

C. 

Ccesar,  his  reform  of  the  Roman  calen- 
dar, 917. 

Calendar,  Julian,  917.    Gregorian,  914... 

Cause  and  effect,  439,  and  note. 

Centre  of  the  earth,  80.  Of  the  sun,  462. 
Of  gravity,  360.  Revolution  about, 
452. 

Centrifugal  force.  Elliptic  form  of  earth 
produced  by,  224.  Illustrated,  225. 
Compared  with  gravity,  229.  Of  a 
body  revolving  on  the  earth's  surface, 
452. 

Ceres,  discovery  of,  505. 

Challis,  Prof.,  506,  note. 

Charts,  celestial,  111.  Construction  of, 
291...     Bremiker'a,  506,  and  note. 

Chinese  records  of  comets,  574.  Of  ir- 
regular stars,  831. 

Chronometers,  how  used  for  determining 
differences  of  longitmle,  255. 

Circle,  arctic  and  antarctic,  94.  Verti- 
cal, 100.  Hour,  106.  Divided,  163. 
Meridian,  174.  Reflecting,  197.  Re- 
peating, 198.     Galactic,  793. 

Clepsydra,  150. 

Clock,  151.  Error  and  rate  of,  how 
found,  253. 

Clouds,  greatest  height  of,  34.  Magel- 
lanic, 892... 

Clusters  of  stars,  864...  Globular,  867. 
Irregular,  869. 

C'Mimalion,  line  of,  155. 

Collimator,  178... 

Coloured  stars,  bol ... 

Colures,  807. 

Comets,  554.     Seen  iu  day  time,  555. 


590.  Tails  of,  556...566.  599.  Ex- 
treme tenuity  of,  558.  General  de- 
scription of,  660.  Motions  of,  and 
described,  561...  Parabolic,  564.  El- 
liptic, 567...  Hyperbolic,  564.  Di- 
mensions of,  565.  Of  Halley,  567... 
Of  CsBsar,  673.  Of  Encke,  676.  Of 
Biela,  579.  Of  Faye,  584.  Of  Lex- 
ell,  585.  Of  De  Vico,  586.  Of  Bror- 
sen,  587.  Of  Peters,  688.  Synopsis 
of  elements  (Appendix).  Increase  of 
visible  dimensions  in  receding  from 
the  sun,  571.  580.  Great,  of  1843, 
589...  Its  supposed  identity  with 
many  others,  694...  Interest  attached 
to  subject,  697.  Cometary  statistics, 
and  conclusions  therefrom,  601. 

Commensurability  (near)  of  mean  mo- 
tions ;  of  Saturn's  satellites,  650.  Of 
Uranus  and  Neptune,  669,  and  note. 
Of  Jupiter  and  Saturn,  720.  Earth 
and  Venus,  726.    Efi"ects  of,  719. 

Compensation  of  disturbances,  how  ef- 
fected, 719.  725. 

Compression  of  terrestrial  spheroid,  221. 

Configurations,  inequalities  depending 
on,  655... 

Conjunctions,  superior  and  inferior,  473. 
Perturbations  chiefly  produced  at,  713. 

Consciousness  of  effect  when  force  is  ex- 
erted, 439. 

Constellations,  60.  301.  How  brought 
into  view  by  change  of  latitude,  62. 
Rising  and  setting  of,  58. 

Copernican  explanation  of  diurnal  mo- 
tion, 76.  Of  apparent  motions  of  sun 
and  planets,  77. 

Correction  of  astronomical  observations, 
324...  8.  Uranographical  summary, 
view  of,  342... 

Culminations,  125.  Upper  and  lower, 
126. 

Cycle,  of  conjunctions  of  disturbing  and 
disturbed  phiHcts,  719.  Metonic,  926. 
Callippic,  ib.  Solar,  921.  Lunar  922. 
Of  indictions,  923. 


D. 


Bay,  solar,  lunar,  and  sidereal,  143. 
Ratio  of  sidereal  to  solar,  305.  909. 
911.  Solar  unequal,  146.  Mean 
ditto  invariable,  908.  Civil  and  astro- 
nomical, 147.     Intercalary,  916. 

Bays  elnpsed  between  principal  chrono- 
logical eras,  926.  Rules  for  reckon- 
ing between  given  dates,  927. 


;t--  ,jf -t,T5^ 


INDEX. 


551 


lower, 


Declination,  105.     How  obtained,  295. 

Definitions,  82... 

Degree  of  meridian,  how  measured,  210... 
Error  admissible  in,  215.  Length  of 
in  various  latitudes,  216.  221. 

Diameters  of  the  earth,  220,  221.  Of 
planets,  synopsis,  Appendix.  [See 
also  each  planet.) 

Dilatation  of  comets  in  receding  from 
the  sun,  578. 

Dione,  548. 

Discs  of  stars,  816. 

Distance  of  the  moon,  403.;  the  sun,  857.; 
fixed  stars,  807.  812... ;  polar,  105. 

Districts,  natural,  in  heavens,  302. 

Disturbing  forces,  nature  of,  609...  Ge- 
neral estimation  of,  611.  Numerical 
values,  612.    Unresolved  in  direction, 

614.  Resolution   of,   in  two   modes, 

615.  618.  Effects  of  each  resolved 
portion,  616...  On  moon,  expressions 
of,  676.  Geomtrical  representations 
of,  676.  717. 

Diurnal  motion  explained,  58.  Paral- 
lax, 339.     Rotation,  144. 

Double  refraction,  202.  Image  micro- 
meter, a  new,  described,  203.  Comet, 
580.     Nebula),  878. 

Double  Stars,  833...  Specimens  of  each 
class,  835.  Orbitual  motion  of,  839. 
Subject  to  Newtonian  attraction,  843. 
Orbits  of  pfJrticular,  843.  Dimen- 
sions of  these  orbits,  844.  848.  Co- 
loured, 851...  Apparent  periods  af- 
fected by  motion  of  light,  863. 

Dove,  his  law  of  temperature,  370. 


S. 


Earth.  Its  motioa  admissible,  15.  Sphe- 
rical form  of,  18.  22...  Optical  effect 
of  its  curvature,  25.  Diurnal  rci;  -ion 
of,  52.  Uniform,  56.  r-ermanencc 
of  its  axis,  57.  Figure  spheroidal, 
219...  Dimensions  of,  220.  Elliptic 
figure  a  result  of  theory,  229.  Tem- 
perature of  surface,  how  maintained, 
366.  Appearance  as  seen  from  moon, 
436.  Velocity  in  its  orbit,  474.  Dis- 
turbance by  Venus,  726. 

Eclipses,  411...  Solar,  420.  Lunar,  421 .. . 
Annular,  425.  Periodic  return  of, 
426.  Number  possible  in  a  year,  426. 
Of  Jupiter's  satellites,  538.  Of  Sa- 
turn's, 549. 

Ecli/itif,  305...  Its  plane  slowly  varia- 
ble, 306.     Cause  of  this  variation  ex- 


plained, 640.    Poles  of,  307.    Limits, 
solar,  412.     Lunar,  427. 

Egyptians,  ancient,  their  chronology,  912. 

Elements  of  a  planet's  orbit,  493,  Varia- 
tions of,  652...  Of  double  star  orbits, 
843.  Synoptic  table  of  planetary, 
&c..  Appendix 

Ellipse,  variable,  of  a  planet,  653.  Mo- 
mentary or  osculating,  654. 

Elliptic  motion  a  consequence  of  gravi- 
tation, 4^6.  Laws  of,  489...  Their 
theoretical  explanation,  491. 

Ellipticity  of  the  earth,  221. 

Elongation,  341.  Greatest,  of  Mercury 
and  Venus,  467. 

Enceladus,  548,  note. 

Encke,  comet  of,  576.  His  hypothesis 
of  the  resistance  of  the  ether,  577. 

Epoch,  one  of  the  elements  of  a  planet's 
orbit,  496.  Its  variation  not  inde- 
pendent, 730.  Variations  incident 
on,  731.  744. 

Equation  of  light,  335.  Of  the  centre, 
375.  Of  time,  379.  Lunar,  452. 
Annual,  of  the  moon,  738. 

Equator,  84. 

Equatorial,  185. 

Equilibrium,  figure  of,  in  a  rotating  body, 
224. 

Equinoctial,  97.     Time,  936. 

Equinox,  293.  303. 

Equinoxes,  precession  of,  312.  Its  ef- 
fects, 313.  In  what  consisting,  31 4... 
Its  physical  cause  explained,  642... 

Eras,  chronological  list  of,  926. 

Irrors,  classification  of,  133.  Instru- 
ijientd,  135...  Their  detection,  140. 
Destruction  of  accidental  ones  by  tak- 
ing means,  137.  Of  clock,  how  ob- 
tained, 293. 

Establishment  of  a  port,  754. 

Ether,  resistance  of,  577. 

Erection  of  moon,  748. 

Excentricities,  stability  of  Lagrange's 
theoi-em  respecting,  701. 

Excentricity  oi  earth's  orbit,  354.  How 
ascertained,  377.  Of  the  moon's,  405. 
Momentary  perturbation  of,  investi- 
gated, 670.  Application  to  lunar 
theory,  688.  Variations  of,  in  orbits 
nearly  circular,  696.  In  excentrio 
orbits,  697.  Permanent  inequalities 
depending  on,  719. 

F. 

Faculce,  338. 

Faye,  comet  of,  584,  and  Appendix. 


ri 


m 


• 


552 


INDEX. 


m 


Flora,  discovery  of,  505. 

Focus,  upper.  Its  momentary  change  of 
place,  G70,  671.  Path  of,  in  virtue  of 
both  elements  of  disturbing  force,  70-1. 
Traced  in  the  case  of  the  moon's  vari- 
ation, 706...  And  parallactic  inequa- 
lity, 712.  Circulation  of,  about  a 
mean  situation  in  planetary  perturba- 
tions, 727. 

Force,  metaphysical  conception  of,  439. 

Forced  vibration,  principle  of,  650. 

Forces,  disturbing.    See  Disturbing  force. 


G. 


Galactic  circle,  793.  Polar  distance, 
ib. 

Galaxy  composed  of  stars,  302.  Sir  W. 
Herschel's  conception  of  its  form  and 
structure,  780.  Distribution  of  stars 
generally  referable  to  it,  7H0.  Its 
course  among  the  constellations,  787... 
Difficulty  of  conceiving  its  real  form, 
792.  Telescopic  analysis  of,  797.  In 
some  directions  unfathomable,  in 
others  not,  798. 

Galle,  Dr.,  506.  Finds  Neptune  in  place 
indicated  by  theory,  768. 

Galloway,  his  researches  on  the  sun's 
proper  motion,  855. 

Gasparis,  Sig.  De,  discovers  a  new  pla- 
net (Appendix). 

Gauging  the  heavers,  793. 

Gay  Lussac,  his  aeronautic  ascent,  32. 

Geocentric  longitude  303.  Place,  371, 
497. 

Geodesical  measurements, — their  nature, 
247... 

Geography,     !i,  205... 

Globular  clusters,  805.  Their  dynami- 
cal stability,  866.  Specimen  list  of, 
867. 

Golden  number,  922. 

Goodricke,  his  discovery  of  variable  stars, 
821... 

OravitatioD,  how  deduced  from  phvno- 
meua,  444...     Klliptic  motion  a  co.i 
sequence  of,  490... 

Gravity,  centre  of,  see  Centre  of  gra- 
vity. 

Gravity  dimini.shed  by  centrifugal  force, 
231.  Mesisure.-j  of,  statical,  234.  Dy 
namical,  235.  Force  of,  oh  the  moon. 
433...  On  bodies  at  nuFfa<;«  of  tlie 
sun,  440.  Of  other  platicts,  see  theit 
names. 

Gregorian  reforim  <A  <iu](,nd»r,  915... 


Ilalley.  His  comet,  567.  First  notices 
proper  motions  of  the  stars,  852. 

Hansen.  His  detection  of  long  inequa- 
lities in  the  moon's  motions,  745... 

Harding  discovers  Juno,  J05. 

Heat,  supply  of,  from  sun  alike  in  sum- 
mer and  winter,  368.  How  kept  up. 
400.  Sun's  expenditure  of,  estimated 
397.  Received  from  the  sun  by  di/- 
ferent  planets,  608.  Endured  by  cc 
mets  in  perihelio,  592. 

Hebe,  discovery  of,  505. 

Heights  above  the  sea,  how  measured 
28().     Mean,  of  the  continents,  289. 

Heliocentric  place,  600. 

Heliometer,  201. 

Henmpheres,  teiTestrial  and  aqueous,  284. 

Herschel,  Sir  Wm.,  discovers  Uranus, 
505,  and  two  satellites  of  Saturn,  548. 
His  method  of  gauging  the  heavens, 
793.  Views  of  the  structure  of  the 
Milky  AVay,  786.  Of  nebular  subsi- 
dence, and  sidereal  aggregation,  869, 
874.  His  catalogues  of  double  stars, 
835.  Discovery  of  their  binary  con- 
nexion, 839.  Of  the  sun's  proper  mo- 
tion, 854.  Classifications  of  nebula, 
868,  879,  note. 

Horizon,  22.  Dip  of,  2^,  195.  Rational 
and  sensible,  74.  Celestial,  98.  Arti- 
iicial,  163. 

Horizontal  point  of  a  mural  circle,  how 
determined,  176... 

Hour  circles,  106 ;  angle,  107 ;  glass, 
150. 

Hyperion,  Appendix,  Saturn's  satellites. 


lapetus,  548. 

Inclination  of  the  moon's  orbit,  400.  Of 
planet's  orbits  disturbed  by  orthogo- 
nal force,  619.  Physical  importance 
of,  as  an  element,  632.  Mtmientary 
variation  of,  estin\iited,  633.  Crite- 
rion of  momentary  increase  or  dimi- 
nution, 635.  Its  changes  periodical 
and  si'lf-correcting,  63().  Application 
to  case  of  the  moon,  638. 

fncUnalions,  stability  of,  Lagrange's  the- 
orem. 639.  Analogous  in  their  per- 
turhHti<*fli*  to  excentricitieji,  699. 

hi/lictiou'.  y2;-. 

twfialily  Ptirallacti*-  of  moon,  712. 
Great.  *0  <>«iyiter  and  Saturn,  "20... 


i:1 


INDEX. 


553 


Tneq^'-alities,  independent  of  excentricity, 
theory  of,  702...     Dependent  on,  719. 
Intercalation,  9]  6. 
/m,  discovery  o*",  506. 
Iron,  meteoric,  888. 


J. 


Julian  Period,  924.  Date,  930.  Re- 
formation, 918. 

Juno,  discovery  of,  505. 

Jupiter,  physical  appearance  and  de- 
scription of,  611.  Ellipticity  of,  612. 
Belts  of,  512.  Gravity  on  surface, 
508.  Satellites  of,  610.  Seen  without 
satellites,  643.  Recommended  as  a 
photometric  standard,  783.  Elements 
of,  &o.  [See  Synoptic  Table,  Appen- 
dix.) 

Jupiter  and  Saturn,  their  mutual  pertur- 
bations, 700,  720... 

K. 

Kater,  his  mode  of  measuring  small  in- 
tervals of  time,  160.  His  collimator, 
178. 

Kepler,  his  laws,  352,  487,  489.  Their 
physical  interpretation,  490... 


Lagging  of  tides,  763. 

Lagrange,  his  theorems  respecting  the 
stability  of  the  planetary  system,  6G9, 
639,  701. 

Laplace  accounts  for  the  secular  accele- 
ration of  the  mc  on,  740. 

Lassell,  his  discovery;  of  the  satellite  of 
Neptune,  624.  Of  an  eighth  satellite 
of  Saturn,  Appendix.  Re-discovers 
two  of  the  satellites  of  Uranus,  551. 

Latitude,  terrestrial,  88.  Parallels  of, 
89.  How  ascertained,  119,  129.  Ro- 
mer's  mode  of  obtaining,  248.  On  a 
spheroid,  247.  Celestial,  308.  Helio- 
centric, how  calculated,  600.  Geo- 
centric, 503. 

Laws  of  nature  how  arrived  at,  139. 
subordinate,  appear  first  in  form  of 
errors,  139.     Kepler's,  362,  487-.. 

Level,  spirit,  176.  Sea,  285.  Strata,  287. 

Leverrier,  606,  607,  767. 

Lcxell,  comet  of,  585. 

Libration  of  the  moon,  435.  Of  apsides, 
694. 

Light,  aberration  of,  331.     Velocity  of, 


331.  How  ascertained,  545.  Equa- 
tion of,  335.  Extinction  of,  in  tra- 
versing space,  798.  Distance  mea- 
sured by  its  motion,  802...  Of  certain 
stars  compared  with  the  sun,  817... 
Efi'ot  of  its  motion  in  altering  appa- 
rent peviod  of  a  double  star,  863. 
Zodiacil,  897. 

Local  time,  252. 

London,  (jentre  of  the  terrestrial  hemi- 
sphere, 284. 

Longitude,  terrestrial,  90.  How  deter- 
mined, 121,  251 ...  By  chronouieiers, 
255.  By  signals,  264.  By  electric 
•telegraph,  262.  By  shooting  stars, 
266.  By  Jupiter's  satellites,  &o.,  266. 
By  lunar  observations,  267...  Celes- 
tial, 308.  Mean  and  true,  375.  He- 
liocentric, 500.  Geocentric,  503.  v)f 
Jupiter's  satellites,  curious  relations 
of,  642. 

Lunation  (synodic  revolution  of  the 
moon),  its  duration,  418. 

M. 

Magellanic  clouds,  892... 

Magnitudes  of  stars,  780...  Common 
and  photometric  scales  of,  780...  and 
Append'!. 

Maps,  geographical,  construction  of,  273. 
Celestial,  290...     Of  the  moon,  437. 

Mars,  phases  of,  484.  Gravity  on  sur- 
face, 508.  Continents  and  seas  of, 
610.     Elements  (Appendix). 

Masses  of  planets  determined  by  their 
satellites,  632.  By  their  mutual  per- 
turbations, 767.  Of  Jupiter's  satel- 
lites, 768.     Of  the  moon,  769. 

Menstrual  equation,  528. 

Mercator's  projections,  283. 

3Iercury,  synodic  revolution  of,  472.  Ve- 
locity in  orbits,  474.  Stationary  points 
of,  476.  Phases,  477.  Greatest  elon- 
gations, 482.  Transits  of,  483.  Hcac 
received  from  sun,  508.  Physical  ap- 
pearance and  description,  509.  Ele- 
ments of  (Appendix). 

Meridian,  terrestrial,  85.  Celestial,  101. 
Line,  87,  190.  Circle,  174.  Marc, 
190.  Arc,  how  ueasured,  213.  Arcs, 
lengths  of,  in  various  latitudes,  216. 

Messier,  his  catalogue  of  nebulu;,  865. 

.Ve/eors,  898.  Periodical,  900...  Heightd 
of,  904. 

Metis,  discovery  of,  505. 

Micrometers,  199. 


i  'I 

i    r 

it 


554 


INDEX. 


Milly  way.     {See  Galaxy,  802.) 

Mimas,  550,  nnd  note. 

Mira  Coti,  820. 

Moon,  her  motion  among  the  stars,  401. 
Distance  of,  403.  Magnitude  and  ho- 
rizontivl  parallax,  404.  Augmenta- 
tion, 4U4.  Her  orbit,  405.  Revolution 
of  nodes,  407.  Apsides,  409.  Oc- 
oultation  of  stars  by,  414.  Phases 
of,  41G.  Brightness  of  surface,  417, 
note.  Redness  in  eclipses,  422.  Phy- 
sical constitution  of,  429...  Destitute 
of  sensible  atmosphere,  431.  Moun- 
tains of,  430.  Climate,  431...  Inha- 
bitimts,  434.  Influence  on  weather, 
432,  and  note.  Rotation  on  axis,  435. 
Appearance  from  earth,  436.  Maps 
and  models  of,  437.  Real  form  of 
orbit  round  the  sun,  452.  Gravity  on 
surface,  508.  Motion  of  her  nodes 
and  change  of  inclination  explained, 
638...  Motion  of  apsides,  676...  Va- 
riation of  excentricity,  688...  Paral- 
lactic inequality,  712.  Annual  equa- 
tion, 738.  Evection,  748.  Variation, 
705...     Tides  produced  by,  751. 

Motion,  apparent  and  real,  15.  Diurnal, 
52.  Parallactic,  68.  Relative  and 
absolute,  78...  Angular,  how  mea- 
sured, 149.  Proper,  of  stars,  852... 
Of  sun,  854. 

Mountains,  their  proportion  to  the  globe, 
29.     Of  the  moon,  430. 

Mowna  lloa,  3;\ 

Mural  circle,  168. 

N. 

Nabonassar,  era  of,  926. 

Nadir,  99. 

Nebula,  classifications  of,  f)68,  879,  note. 
Law  of  distribution,  868.  Resolvable, 
870.  Elliptic,  873.  Of  Andromeda, 
874.  Annular,  875.  Planetary,  876. 
Colourr.l,  ib.  ^ouble,  878.  Of  sub- 
regulai  'orms,  881,  882.  Irregular, 
883.  Of  Orion,  885.  Of  .  -go,  887. 
Of  Sagittarius,  888.    Of  Cyguus,  891. 

Nebular  hypothesis,  872. 

Nebulous  matter,  871.     Stars,  880. 

Neptune,  discovery  of,  506,  768.  Pertur- 
bations produced  on  Uranus  by,  ana- 
lysed, 765...  Place  indicated  by  the- 
ory, 767.  Elements  of,  771...  Per- 
turbing forces  of,  on  Uranus,  geome- 
trically exhibited,  773.  Their  effects. 
771... 


Newton,  his  theory  of  gravitation,  490... 
et  pastim. 

Nodes  of  the  sun's  equator,  890.  Of  the 
moon's  orbit,  407.  Passage  of  pla- 
nets through,  460.  Of  planetary  or- 
bits, 495.  Perturbation  of,  620... 
Criterion  of  their  advance  or  recess, 
622.  Recede  on  the  disturbing  orbit, 
624...  Motion  of  the  moon's  theory 
of,  638.  Analogy  of  their  variations 
to  those  of  perihelia,  699. 

Nomenclature  of  Saturn's  satellites,  548, 
note. 

Nonagesimal  point,  how  found,  310. 

Normal  disturbing  force  and  its  effects, 
618.  Action  on  excentricity  and  pe- 
rihelion, 673.  Action  on  lunar  ap- 
sides, 676.  Of  Neptune  on  Uranus, 
its  effects,  775. 

Nubeculce,  major  and  minor,  892... 

Number,  golden,  922. 

Nutation,  in  what  consisting,  321.  Pe- 
riod, 322.  Common  to  all  celestial 
bodies,  323.  Explained  an  physical 
principles,  648. 

0. 

Obliquity  of  ecliptic,  303.  Pioduces  the 
variations  of  season,  302.  Slowly 
diminishing,  and  why,  640. 

Observation,  astronomical,  its  peculiari- 
ties, 138. 

Occultation,  perpetual,  circle  of,  113. 
Of  a  star  by  the  moon,  413...  Of  Ju- 
piter's satellites  by  the  body,  541. 
Of  Saturn's,  549. 

Olbers  discovers  Pallas  and  Vesta,  505. 
His  hypothesis  of  the  partial  opacity 
of  space,  798. 

Opacity,  partial,  of  space,  798. 

Oscillations,  forced,  principle  of,  650. 

Orbits  of  planets,  their  elements  (Ap- 
pendix) of  double  stars,  843.  Of 
comets.     (See  Comets.) 

Orthogonal  disturbing  force,  and  its  ef- 
fects, 616,  619. 

Orthographic  projection,  280. 


PaUtzch  discovers  the  variability  of  Al- 
gol. 821. 

Palliu,  discovery  of,  506. 

Parallactic  instrument,  18  .  Inequality 
of  the  moon,  712  O;  planets,  713. 
Unit  of  sidereal  u  uiuces,  804.  Mo- 
tion, 68. 


INDEX. 


555 


ritation,  490... 

',  890.  Of  the 
nasage  of  pla- 
'  planetary  or- 
lon  of,  620... 
nee  or  recess, 
sturbing  orbit, 
moon's  theory 
leir  variatioua 
99. 
satellites,  548, 

.und,  310. 
md  its  effects, 
tricity  and  pe- 
on lunar  ap- 
16  on  Uranus, 

)r,  892... 

ing,  321.  Pe- 
.0  all  celestial 
d   an  physical 


Pioduces  the 
302.  Slowly 
340. 

its  peculiari- 

ircle  of,  113, 
113...  OfJu- 
le  body,  5-11. 

id  Vesta,  505. 
)artial  opacity 

798. 
pie  of,  650. 

lementa  (Ap- 
rs.  848.      Of 

e,  and  its  ef- 

BO. 


[ability  of  Al- 


Inequality 
planets,  718. 
;s,  804.     Mo- 


Pjrallax,  70.  Geocentrio  or  diurnal, 
839.  Heliocentric,  841.  Horizontal, 
855.  Of  the  moon,  404.  Of  the  sun, 
857,  479,  481.  Annual,  of  stars,  800. 
How  investigated,  805...  Of  particu- 
lar Htars,  812,  818,  816.  Systematic, 
862. 

Peak  of  Teneriffe,  82. 

Pendulum-cloek,  89.  A  meoisure  of  gra- 
vity, .235. 

Penumbra,  420. 

Perigee  of  moon,  406. 

Perihelia  and  excentrioities,  theory  of, 
670... 

Perihelion,  368.  Longitude  of,  495. 
Passage,  496.  Heat  endured  by  co- 
mets in,  592. 

Period,  Julian,  924.     Of  planets  (App.). 

Periodic  time  of  a  body  revolving  at  the 
earth's  surface,  442.  Of  planets,  hovr 
ascertained,  486.  Law  of,  48.  Of  a 
disturbed  planet  permanently  altered, 
734... 

Periodical  stars,  820...     List  of,  825. 

Perspective,  celestial,  114. 

Perturbations,  602... 

Peters,  his  researches  on  parallax,  815. 

Phases  of  the  moon  explained,  416.  Of 
Mercury  and  Venus,  465,  477.  Of 
superior  planets,  484. 

Photometric  scale  of  star  magnitudes,  780. 

Piazzi  discovers  Cere.s,  505. 

Pigott,  variable  stars  discovered  by, 
824. 

Places,  mean  and  true,  374.  Geometric 
and  heliocentric,  871,  497. 

Planetary  ..obulae,  876. 

Planet',  456.  Zodiacal  uuJ  u>ra-zodia- 
cal,  457.  Apparent  motions,  459. 
Stations  and  retrogradations,  459. 
Referpuce  to  sun  as  their  centre,  4G2. 
Community  of  nature  with  the  earth, 
4C3.  Apparent  diameters  of,  464. 
Phases  of.  405  Inferior  and  superior, 
407.  Transits  of  {see  Transit)  Mo- 
tions explained,  408.  Distances,  how 
concluded,  471.  Periods,  how  lound, 
472.  Synodical  revolution,  472.  Su- 
perior, tk*»T  (Stations  and  retrograiJa- 
ti'/fis,  4fct/.  Magnitude  of  orbitH,  low 
coflchided,  485.  Elements  of,  495. 
(A>t'  Appenrlix  for  Synoptic  Table.) 
Densities,  508.  Physical  peculiarities, 
6:c.,  509...  llluetration  of  their  rela- 
tive a'vMx  and  distances,  520. 

Plundiiiiuur,  his  calculations  respectiug 
(he  double  comei  of  Biela,  583. 


Pleiades,  805.  Assigned  by  Miidler  as 
the  central  point  of  the  sidereal  sys- 
tem, 861. 

Plumb-line,  direction  of,  23.  Use  of,  in 
observation,  176. 

Polar  distance,  105.  Point,  on  a  mural 
circle,  170,  172. 

Poles,  83.     Of  ecliptic,  807. 

Pole-star,  59.  Useful  for  finding  the 
latitude,  171.  Not  always  the  same, 
818.  What,  at  epoch  of  the  building 
of  the  pyramids,  319. 

Pores  of  the  sun's  surface,  387. 

Position,  angle  of,  204.     Micrometer,  ib. 

Precession  of  the  equinoxes,  312.  In  what 
consisting,  314...  Effects,  313.  Phy- 
sical explanation,  642. 

Prcesejpe,  Canori,  865. 

Priming  and  lagging  of  tides,  753. 

Principle  of  areas,  490.  Of  forced  vibra- 
tions, 650.  Of  repetition,  198.  Of 
conservation  of  vis  viva,  663.  Of  col- 
limation,  178. 

Problem  of  three  bodies,  008. 

Problems  in  plane  astronomy,  127... 
309... 

Projection  of  a  star  on  the  moon's  limb, 
414,  note. 

Projections  of  the  sphere,  280... 

Proper  motions  of  the  stary,  862.  Of  tbo 
sun,  853. 

Pyrar'ids,  319. 

R. 

Radial  disturbing  force,  615... 

Radiation,  solar,  on  planets,  508.  On 
comets,  692. 

Rate  of  clock,  how  obtained,  293. 

Reading  off,  met'.\ods  of,  105. 

Rejiexion,  observations  by,  173. 

Refraction,  38.  Astronomical  and  its 
effects,  39,  40.  Measure  of,  and  law 
of  variation,  43.  How  detected  by 
observation,  142.  Terrestrial,  44. 
How  best  investigated,  191. 

Repetition,  principle  of,  198, 

Resistance  of  ether,  577. 

Retrogradations  of  planets,  459.  Of 
nodes.     [See  Nodes.) 

Rhea,  548,  note. 

Right  ascension,  108,  How  determined, 
293. 

Rings  of  Saturn,  dimensions  of,  514. 
Pheuomena  of  their  disappearance, 
615...  Equilibriuui  of,  518...  rWui« 
tiple,  521,  and  Appendix     Appear 


II 


656 


INDEX. 


anoe  of  from  Saturn,  522.     Attraction 

of  on  a  point  within,  786,  note. 
Rittenhouae,  hia  principle  of  collimation, 

178. 
Roase,  Earl  of,  his  great  reflector,  870, 

882. 
Rotation,  diurnal,  68.     Parallactic,  68. 

Of  planets,  609...     Of  Jupiter,  512. 

Of  fixed  stars  on  their  axes,  820. 


S. 


Saros,  426. 

SalelHtes,  of  Jupiter,  511.  Of  Saturn, 
518,  547.  Discovery  of  an  eighth 
(Appendix).  Of  Uranus,  523,  662. 
Of  Neptune,  624,  563.  Used  to  de- 
termine masses  of  their  primaries, 
632.  Obey  Kepler's  laws,  633.  Eclipses 
of  Jupiter's,  535...  Other  phasnomena 
of,  540.  Their  dimensions  and  masses, 
540.  Discovery,  544.  Velocity  of 
light  ascertained  from,  545. 

Saturn,  remarkable  deficiency  of  density, 
508.  Rings  of,  514.  Physical  descrip- 
tion of,  614.  Satellites  of,  547,  and 
Appendix.  (^See  also  elements  in  Ap- 
pendix.) 

S(0,  proportion  of  its  depth  to  radius  of 
the  glube,  31.  Its  action  in  modelling 
the  external  form  of  the  earth,  227. 

Seasons  explained,  302...  Temperature 
of,  366. 

Sector,  zenith,  192. 

Secular  variations,  655. 

Selenography,  437. 

Sextant,  l'J3... 

Shadow,  dimensions  of  the  earth's,  422, 
428.  Cast  by  Venus,  467.  Of  Jupi- 
ter's satellites  seen  on  disc,  540. 

Shootint/  stars  used  for  finding  longi- 
tudes, 265.  Periodical,  900.  (&e 
Meteors. ) 

Sidereal  time,  110,  910.  Year.  {See 
Year.)     Day.     {See  Day.) 

Si[/ns  of  zodiac,  880. 

Sinnti,  its  parallax  and  absolute  light, 
818. 

Solar  cycle,  921. 

Sphere,  95.  Projections  of,  280.  Attrac- 
tion of,  735,  note. 

Spheroidal  form  of  Earth  [see  Earth)  pro- 
duces inequalities  in  the  moon's  mo- 
tion, 749. 

Spots  on  Sun,  389... 

Stars  visible  by  duy,  61.  Fixed,  777... 
Their    apparent    magnitudes,    778... 


Comparison  by  an  astromcter,  788. 
Law  of  distribution  over  heavens, 
785...  alike  in  either  hemisphere,  794. 
Parallax  of  certain,   815.     Discs  of, 

816.  Real   size  and   absolute  light, 

817,  Periodical,  820.  Temporary, 
827.  Irregular,  830.  Missing,  832. 
Double,  883...  Coloured,  851,  and 
note.  Proper  motions  of,  852.  Irre- 
gularities in  motions  not  verified,  859. 
Clusters  of,  864...  Nebulous,  879... 
Nebulous-double,  880. 

Stationary  points  of  planets,  469.  How 
determined,  475.  Of  Mercury  and 
Venus,  476. 

Stereographic  projection,  281. 

Stones,  meteoric,  898.  Great  shower  of, 
ib, 

Strtive,  his  researches  on  the  law  of  dis- 
tribution of  stars,  793.  Discovery  of 
parallax  of  a  Lyrao,  818,  Catalogue 
and  observation  of  double  stars,  835. 

Struve,  Otto,  his  researches  on  proper 
motions,  854. 

Style,  old  and  new,  932. 

Suji,  oval  shape  and  great  size  on  hori- 
zon explained,  47.  Apparent  motion 
not  uniform,  34.  Orbit  elliptic,  349. 
Greatest  and  least  distances,  350. 
Actual  distance,  357.  Magnitude, 
358.  Rotation  on  axis,  359,  390. 
Mass,  360.  Physical  constitution, 
3S6.  Spots,  ib...  Situation  of  its 
ciiuator,  390...  Maculiferous  zones 
of,  303.  Atmosphere,  395.  Tempe- 
rature, 896.  Expenditure  of  heat, 
397.  Eclipses,  420.  Density  of,  447. 
Natural  centre  of  planetary  system, 
462.  Distance,  how  determined,  479. 
Its  size  illustrated,  526.  Action  in 
producing  tides,  751.  Proper  motion 
of,  854...  Absolute  velocity  of  in 
space,  858.  Central,  speculations  on, 
861. 

Sunsets,  two,  witnessed  in  one  day,  26. 

Survey,  trigonometrical,  nature  of,  274. 

Synodic  revolution,  418.  Of  sun  and 
moon,  ib. 

T. 

Tangential  force  and  its  effects,  618. 
Momentary  action  on  perihelia,  673. 
Wholly  influential  on  velocity,  600. 
Produces  variations  of  axis,  iO... 
Doubles  the  rate  of  advance  of  lunar 
apsides,  686.  Of  Neptune  on  Uranus, 
and  its  efiects,  774. 


INDEX. 


657 


Telescope,  154.  Its  appHoAtion  to  astro- 
nomical instruments,  117. 

Telescopic  sights,  invention  of,  158,  note. 

Temperature  of  earth's  surface  at  differ- 
ent seasons,  866.  In  South  Africa 
and  Australia,  869.     Of  the  sun,  896. 

Teihys,  648,  note. 

Theodolite,  192.  Its  use  in  surrcying, 
276. 

Theory  of  instrumental  errors,  141,,  Of 
gravitation,  490...  Of  nebulous  sub- 
sidence and  sidereal  aggregation,  872. 

Tides,  a  system  of  forced  oscillations, 
051.  Explained,  750...  Priming  and 
lagging  of,  763.  Periodical  inequali- 
ties of,  755.  Instances  of  very  high, 
756. 

Time,  sidereal,  110,  327,  911.  Local, 
129,  162.  Measures  angular  motion, 
149.  How  itself  measured,  150... 
Very  small  intervals  of,  160.  Equi- 
noctial, 257,  925...  Measures,  units, 
and  reelioning  of,  906... 

Titan,  648,  note. 

Titius,  Prof.,  his  law  of  planetary  diis"- 
tances,  606,  note. 

Trade  winds,  239... 

Transit  instrument,  150... 

Transits  of  atnra,  152.  Of  planets  across 
the  sun,  467.  Of  Venus,  479...  Mer- 
cury, 483.  Of  Jupiter's  satellites 
across  disc,  540.  Of  their  shadows, 
549. 

Transparency  of  space,  supposed  by  01- 
bers  imperfect,  798. 

Transversal  disturbing  force,  and  its 
effects,  615... 

Trigonometrical  survey,  274. 

Tropics,  93,  880. 

Twilight,  44. 


U. 


Umbra  in  eclipses,  420.  Of  Jupiter,  538. 

Uranoyraphy,  111,  800. 

Uranographical  corrections,  342...  Pro- 
blems, 127...  309... 

Uranus,  discovery  of,  505.  Heat  received 
from  sun  by,  508.     Physical  descrip- 


tion of,  523.  Satellites  of,  551.  Per- 
turbations of  by  Neptune,  780...  Old 
observations  of,  760. 


Vanishing  point  of  parallel  lines,  116. 
Lino  of  parallel  planes,  117. 

Variation  of  the  moon  explained,  705... 

Variations  of  elements,  663,  Periodical 
and  secular,  665.  Incident  on  the 
epoch,  731. 

Velocicy,  angular,  of  sun  not  uniform, 
360.  Muear,  of  sun  not  uniform,  861. 
Of  planets,  Mercury,  Venus,  and  Earth, 
474.  Of  'ight,  646.  Of  shooting 
stars,  899,  'j04. 

Venus,  synodic  revolution  of,  472.  Sta- 
tionary points,  476.  Velocity  of,  474. 
Phases,  477.  Points  of  greatest 
brightness,  478.  Transits  of,  479. 
Physical  description  and  appearance, 
609.  Inequality  in  earth's  motion 
produced  by,  726.  In  that  of  the 
moon,  743... 

Verttier,  97. 

Vertical,  prime,  102.     Circles,  100. 

Vesta,  discovery  of,  506. 

W. 

Weight  of  bodies  in  different  latitudes, 
822.  Of  tt  body  on  the  moon,  608. 
On  the  sun,  460. 

Winds,  trade,  24.0... 


Year,  sidereal,  806.  Tropical,  383. 
Anomalistic,  884,  and  day  incommen- 
surable, 918.  Leap,  914.  Of  confu- 
sion, 917.  Beginning  of,  in  England, 
changed,  982. 


Zenith,  99.     Sector,  192. 

Zodiac,  305. 

Zodiacal  light,  899. 

Zones  of  climate  and  latitude,  882. 


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